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Giambattista College Physics Chapter 18


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©2020 McGraw-Hill Education

Chapter 18: Electric Current and Circuits

18.1 Electric Current.

18.2 Emf and Circuits.

18.3 Microscopic View of Current in a Metal: The Free-Electron Model.

18.4 Resistance and Resistivity.

18.5 Kirchhoff’s Rules.

18.6 Series and Parallel Circuits.

18.7 Circuit Analysis Using Kirchhoff’s Rules.

18.8 Power and Energy in Circuits.

18.9 Measuring Currents and Voltages.

18.10 RC Circuits.

18.11 Electrical Safety.


18.1 Electric Current

A net flow of charge is called an electric current .

The current (symbol I ) is defined as the net amount of charge passing per unit time through an area perpendicular to the flow direction.

Recall this from CHAP 16:

Neutral

Charge moves(not really)

Thing 1 Thing 2

- +





What’s really going on...In this picture, five electrons (so a charge of ) move past surface A TO THE LEFT. So this is the effect of a positive current to the RIGHT.

8⋅10−19C


Definition of Current

The SI unit of current, equal to one coulomb per second, is the ampere (A).

qI

t

1 C 1 A s


Example 18.1

Two wires of cross-sectional area 1.6 mm2 connect the terminals of a battery to the circuitry in a clock. During a time interval of 0.040 s, 5.0 × 1014 electrons move to the right through a cross section of one of the wires. (Actually, electrons pass through the cross section in both directions; the number that cross to the right is 5.0 × 1014 more than the number that cross to the left.)

What is the magnitude and direction of the current in the wire?


Example 18.1 Strategy

Current is the rate of flow of charge.

We are given the number N of electrons; multiplying by the elementary charge e gives the magnitude of moving charge Δq.


Example 18.1 Solution

14 19 5

5

5.0 10 1.60 10 C 8.0 10 C

8.0 10 C 0.0020 A 2.0 mA

0.040 s

q Ne

qI

t


Electric Current in Liquids and Gases

Electric currents can exist in liquids and gases as well as in solid conductors.

In an ionic solution, both positive and negative charges contribute to the current by moving in opposite directions.

Since positive and negative charges are moving in opposite directions, they both contribute to current in the same direction.


Application : Current in Neon Signs and Fluorescent Lights


18.2 Emf and Circuits

To maintain a current in a conducting wire, we need to maintain a potential difference between the ends of the wire.

One way to do that is to connect the ends of the wire to the terminals of a battery (one end to each of the two terminals).

An ideal battery maintains a constant potential difference between its terminals, regardless of how fast it must pump charge to do so. An ideal battery is analogous to an ideal water pump that maintains a constant pressure difference between intake and output regardless of the volume flow rate.


Circuit Symbols for a Battery

Of the two vertical lines, the long line represents the terminal at higher potential and the short line represents the terminal at lower potential.


Electromotive Force

The potential difference maintained by an ideal battery is called the battery’s emf (symbol ℰ).

Emf originally stood for electromotive force , but emf is not a measure of the force applied to a charge or to a collection of charges; emf cannot be expressed in newtons.

Rather, emf is measured in units of potential (volts) and is a measure of the work done by the battery per unit charge. To avoid this confusion, we just write “emf” (pronounced ee-em-ef).


Work Done by an Ideal Battery

If the amount of charge pumped by an ideal battery of emf ℰ is q, then the work done by the battery is.

Work done by an ideal battery:

W=ξ q


Emf in an Electric Circuit



MinnesotaGulf

MississippiRiver


Batteries

Batteries come with various emfs (12 V, 9 V, 1.5 V, etc.) as well as in various sizes. The size of a battery does not determine its emf. (Notice that most batteries are a multiple of 1.5 V)

Common battery sizes AAA, AA, A, C, and D all provide the same emf (1.5 V). However, the larger batteries have a larger quantity of the chemicals and thus store more chemical energy.

A larger battery can supply more energy by pumping more charge than a smaller one, even though the two do the same amount of work per unit charge.


EXAMPLE: Samsung S21 Ultra Battery


According to the company, the Samsung S21 Ultracell phone has a battery “size” of 5000 mAh. What is this? Check the units:

mAh = milli-Amp-hour =

So this “size” is not a physical size, but how much charge the battery can “lift” to the higher voltage. (But it DOES correlate with the physical size).

(5 A)(3600 s)=18⋅103C



So this “size” is not a physical size, but how much charge the battery can “lift” to the higher voltage.

Cell phone batteries usually have an “emf” of 3.7 V (note this breaks my “multiple of 1.5 V” rule – cell phone batteries use a different chemistry and so have a different emf)

So the S21 battery can raise 18,000 C of charge to a potential 3.7 V higher than the “ground” of the battery. So the total work the battery can do is:

(about the same as 2 mL = 2 milliliters of gasoline, which is about the volume of three nickels)

W=ξ q=(3.7V )(18⋅103C)=67 kJ



So the total work the battery can do is:

According to Tom’s Guide, the S21 Ultra battery lasts about 10 hours (= 36,000 sec), so the power provided by the battery is:

W=ξ q=(3.7V )(18⋅103C)=67 kJ

P= energytime

= 67 kJ36 ksec

=1.9W


Circuits

Current does not get “used up” in the light bulb any more than water gets used up in the radiator.

©Charles D. Winters/Science Source


Circuits


Current is a flow of charge, and as far as we know, charge is conserved (never created or destroyed).

Energy is conserved – an electrical circuit is generally a way to convert electrical energy into some other form:Light (from a bulb), Motion (an electric fan), Heat (an electric stove)

Electrical energy can come from:Light (a photovoltaic), Motion (a windmill or dam), Heat (a solar thermal or thermoelectric device)Civilization is about converting energy from one form to another.



Direct Current

In this chapter, we consider only circuits in which the current in any branch always moves in the same direction—a direct current (dc) circuit.

In Chapter 21, we study alternating current (ac) circuits, in which the currents periodically reverse direction.


18.3 Microscopic View of Current in a Metal: The Free-Electron Model

The electrons have a nonzero average velocity called the drift velocity, vD. The magnitude of the drift velocity (the drift speed ) is much smaller than the instantaneous speeds of the electrons.


The Drift Velocity is Constant

An electron has a uniform acceleration between collisions, but every collision sends it off in some new direction with a different speed. Each collision between an electron and an ion is an opportunity for the electron to transfer some of its kinetic energy to the ion.

The net result is that the drift velocity is constant, and energy is transferred from the electrons to the ions at a constant rate.


Relationship Between Current and Drift Velocity 1

The number of electrons in the volume is N = nAvDΔt ; the magnitude of the charge is.

DQ Ne neA t



Therefore, the magnitude of the current in the wire is.

Remember that, since electrons carry negative charge, the direction of current flow is opposite the direction of motion of the electrons. The electric force on the electrons is opposite the electric field, so the current is in the direction of the electric field in the wire.

DQ Ne neA t

D

QI neA

t



The equation

can be generalized to systems in which the current carriers are not necessarily electrons, simply by replacing e with the charge of the carriers:

The quantities v+ and v− are drift speeds —both are positive.

D

QI neA

t

I n eA n eA


Example 18.2

A #12 gauge copper wire, commonly used in household wiring, has a diameter of 2.053 mm. There are 8.00 × 1028 conduction electrons per cubic meter in copper.

If the wire carries a constant dc current of 5.00 A, what is the drift speed of the electrons?



2 21

4A r d

228 3 19 3

4 1

5.00 A1

8.00 10 m 1.602 10 C 2.053 10 m4

1.179 10 m s 0.118 mm/s

D

I

neA

SNAIL: ~3-13 mm/s TORTOISE: ~ 110 mm/s


18.4 Resistance and Resistivity

Suppose we maintain a potential difference across the ends of a conductor. How does the current I that flows through the conductor depend on the potential difference ΔV across the conductor?

For many conductors, the I is proportional to ΔV.


Ohm’s Law Equation

Ohm’s law is not a universal law of physics like the conservation laws. THEN WHY CALL IT THAT!

It does not apply at all to some materials, whereas even materials that obey Ohm’s law equation for a wide range of potential differences fail to do so when Δ V becomes too large.

Any homogeneous material follows Ohm’s law equation for some range of potential differences; metals that are good conductors follow Ohm’s law equation over a wide range of potential differences.

I V



TL; DR

Ohm’s law equation works when it works and not when it doesn’t – it will ALWAYS work over a limited range (big or small) and NEVER works for all conditions.

I V


Resistance

The electrical resistance R is defined to be the ratio of the potential difference (or voltage ) ΔV across a conductor to the current I through the material.

Definition of resistance (NOT Ohm’s law equation):

In SI units, electrical resistance is measured in ohms (symbol Ω, the Greek capital omega), defined as.

VR

I

1 1 V/A


Resistance

Think of “resistance” like a drag force for moving charges – in the same way that water “resists” you walking way more than air “resists” you walking, some materials “resist” the passage of charged particles more or less – Essentially, it is the number of things that the electron (or whatever) collides with that prevent it from moving to lower potential energy in the way that air drag slows down a falling object as the falling object collides with air molecules.


Ohmic and Nonohmic Conductors

An ohmic conductor—one that follows Ohm’s law equation—has a resistance that is constant, regardless of the potential difference applied.

For an ohmic conductor, a graph of current versus potential difference is a straight line through the origin with slope 1/R.

For some nonohmic systems, the graph of I versus ΔV may be dramatically nonlinear.


Ohmic Conductors: Example


Nonohmic Conductors: Example

Access the text alternative for these images


Resistivity 1

Resistance depends on size and shape. We expect a long wire to have higher resistance than a short one (in a longer wire, there are more things to bang into – imagine a long road, which has more stop signs to slow you down.) and a thicker wire to have a lower resistance than a thin one (like a multi-lane road which has more paths to follow to lower potential energy).

The electrical resistance of a conductor of length L and cross-sectional area A can be written:

Resistance = [properties of material] x [geometry of resistor]

LR

A


Resistivity 2

The constant of proportionality ρ (Greek letter rho), which is an intrinsic characteristic of a particular material at a particular temperature, is called the resistivity of the material.

The SI unit for resistivity is Ω·m. (Be prepared to argue why that MUST be true)

The inverse of resistivity is called conductivity [SI units (Ω·m) −1].

LR

A


Resistivities for Selected Materials

Table 18.1 Resistivities and Temperature Coefficients at 20°C.

ρ (Ω·m) α (°C–1) ρ (Ω·m) α (°C–1)

Conductors Semiconductors (pure)

Silver 1.59 × 10−8 3.8 × 10−3 Carbon 3.5 × 10−5 −0.5 × 10–3

Copper 1.67 × 10−8 4.05 × 10−3 Germanium 0.6 −50 × 10–3

Gold 2.35 × 10−8 3.4 × 10−3 Silicon 2300 −70 × 10–3

Aluminum 2.65 × 10−8 3.9 × 10−3

Tungsten 5.40 × 10−8 4.50 × 10−3

Iron 9.71 × 10−8 5.0 × 10−3 Insulators

Platinum 10.6 × 10−8 3.64 × 10−3 Wood 108 − 1011

Lead 21 × 10−8 3.9 × 10−3 Glass 1010 − 1014

Manganin 44 × 10−8 0.002 × 10−3 Rubber (hard) 1013 − 1016

Constantan 49 × 10−8 0.002 × 10−3 Lucite > 1013

Mercury 96 × 10−8 0.89 × 10−3 Teflon > 1013

Nichrome 108 × 10−8 0.4 × 10−3 Quartz (fused) > 1016


Resistivity of Water

The resistivity of water depends strongly on the concentration of ions.

Pure water contains only the ions produced by self-ionization (H2O ↔H+ + OH− ). As a result, pure water is an insulator. (ONLY BECAUSE there is such a small number of ions)

Even a small amount of dissolved minerals dramatically lowers the resistivity. The resistivity is so sensitive to the concentration of impurities that resistivity measurements are used to determine water purity.


Example 18.3

(a) A 30.0-m-long extension cord is made from two #19-gauge copper wires. (The wires carry currents of equal magnitude in opposite directions.) What is the resistance of each wire at 20.0°C? The diameter of #19-gauge wire is 0.912 mm.

(b) If the copper wire is to be replaced by an aluminum wire of the same length, what is the minimum diameter so that the new wire has a resistance no greater than the old?


Example 18.3 Solution 1

(a)

8

22 4 7 2

1.67 10 m

1 19.12 10 m 6.533 10 m

4 4A d

8

7 2

1.67 10 m 30.0 m

6.533 10 m0.767

LR

A



(b)a c

a c

2 2a c

2 2a c c a

1 14 4

R R

L L

d d

d d

8a

a c 8c

2.65 10 m0.912 mm 1.15 mm

1.67 10 md d


Resistors 1

A resistor is a circuit element designed to have a known resistance.

In circuit analysis, it is customary to write the relationship between voltage and current for a resistor as V = IR. (which is good, since that’s the definition…..)

Remember that V actually stands for the potential difference between the ends of the resistor even though the symbol Δ is omitted.

©Artit Thongchuea/Shutterstock


Resistors 2

Current in a resistor flows in the direction of the electric field, which points from higher to lower potential.

Therefore, if you move across a resistor in the direction of current flow, the voltage drops by an amount IR.

In a circuit diagram, the symbol.

represents a resistor or any other device in a circuit that dissipates electric energy.

A straight line in a circuit diagram represents a conducting wire with negligible resistance.


Internal Resistance of a Battery 1



When the current through a source of emf is zero, the terminal voltage—the potential difference between its terminals—is equal to the emf.

When the source supplies current to a load (a light bulb, a toaster, or any other device that uses electric energy), its terminal voltage is less than the emf; there is a voltage drop due to the internal resistance of the source.

If the current is I and the internal resistance is r, then the voltage drop across the internal resistance is Ir and the terminal voltage is

V B=ξ −I r


18.5 Kirchhoff’s Rules 1

Two rules, developed by Gustav Kirchhoff (1824 to 1887), are essential in circuit analysis.

Kirchhoff’s junction rule states that the sum of the currents that flow into a junction - any electric connection - must equal the sum of the currents that flow out of the same junction.

The junction rule is a consequence of the law equation of conservation of charge. Since charge does not continually build up at a junction, the net rate of flow of charge into the junction must be zero.


Kirchhoff’s Junction Rule

outin 0I I



Kirchhoff’s loop rule is an expression of energy conservation applied to changes in potential in a circuit.

Recall that the electric potential must have a unique value at any point; the potential at a point cannot depend on the path one takes to arrive at that point.

Therefore, if a closed path is followed in a circuit, beginning and ending at the same point, the algebraic sum of the potential changes must be zero.


Kirchhoff’s Loop Rule

For any path in a circuit that starts and ends at the same point. (Potential rises are positive; potential drops are negative.)

If you follow a path through a resistor going in the same direction as the current, the potential drops (ΔV = − IR). If your path takes you through a resistor in a direction opposite to the current (“upstream”), the potential rises (ΔV = + IR). For an emf, the potential drops if you move from the positive terminal to the negative (ΔV = −ℰ); it rises if you move from the negative to the positive (ΔV = +ℰ).

0V


Kirchhoff’s “Rules”

Loop “rule” says that a charge can gain or lose electrical energy in a circuit, but that all GAINS are balanced by LOSSES – conserved!(Technically, this is electrical energy gained and lost – it is really only converted – a battery converts chemical energy to electrical energy and a resistor converts electrical energy into sound/light/heat/whatever)

Junction “rule” says that you can’t create or destroy charge – conserved!

These “rules” come from conservation laws (and Ohm’s “law” is just a rule of thumb that works when it works...)


18.6 Series and Parallel Circuits

When one or more electric devices are wired so that the same current flows through each one, the devices are said to be wired in series.


Resistors in Series 1

The circuit shows two resistors in series. The straight lines represent wires, which we assume to have negligible resistance.

Negligible resistance means negligible voltage drop (V = IR), so points connected by wires of negligible resistance are at the same potential.

The junction rule, applied to any of the points A – D, tells us that the same current flows through the emf and the two resistors.



Let’s apply the loop rule to a clockwise loop DABCD . From D to A we move from the negative terminal to the positive terminal of the emf, so ΔV = +1.5 V. Since we move around the loop with the current, the potential drops as we move across each resistor.

1 21 5 V 0IR IR

1 2 1 5 VI R R



1 2Req R R

eq 1 2 1 5 VIR I R R



For any number N of resistors connected in series,

Note that the equivalent resistance for two or more resistors in series is larger than any of the resistances.

eq i 1 2 NR R R R R


Emfs in Series 1

In many devices, batteries are connected in series with the positive terminal of one connected to the negative terminal of the next.

This provides a larger emf than a single battery can.


Emfs in Series 2

The emfs of batteries connected in this way are added just as series resistances are added. However, there is a disadvantage in connecting batteries in series: the internal resistance is larger because the internal resistances are in series as well.


Emfs in Series 3

Sources can be connected in series with the emfs in opposition. A common use for such a circuit is in a battery charger. In the figure, as we move from point C to B to A, the potential decreases by ℰ2 and then increases by ℰ1, so the net emf is ℰ1 − ℰ2.



Capacitors in Series 1

The figure shows two capacitors connected in series. Although no charges can move through the dielectric of a capacitor from one plate to the other, the instantaneous currents I that flow onto one plate and from the other must be equal.

Why? The two plates of a capacitor always have charges of equal magnitudes and opposite signs. Therefore, the magnitudes of the charges on the two plates must change at the same rate . The rate of change of the charge is equal to the current. Viewed from the outside, the capacitor behaves as if a current I flows through it.



We want to find the equivalent capacitance Ceq that would store the same amount of charge as each of the series capacitors for the same applied voltage.

With the switch closed, the emf pumps charge so that the potential difference between points A and B is equal to the emf. The capacitors are fully charged and the current goes to zero. From Kirchhoff’s loop rule,


ξ −V 1−V 2=0



The equivalent capacitance is defined by ℰ = Q/ Ceq.

1 21 2

andQ Q

V VC C

eq 1 2

eq 1 2

0

1 1 1

Q Q Q

C C C

C C C

ξ −V 1−V 2=0



For N capacitors connected in series,

Note that the equivalent capacitor stores the same magnitude of charge as each of the capacitors it replaces.

eq 1 2

1 1 1 1 1

i NC C C C C


Resistors in Parallel 1

When one or more electrical devices are wired so that the potential difference across them is the same, the devices are said to be wired in parallel.



In the figure, an emf is connected to three resistors in parallel with each other. The left side of each resistor is at the same potential since they are all connected by wires of negligible resistance.

Likewise, the right side of each resistor is at the same potential. Thus, there is a common potential difference across the three resistors.




Applying the junction rule to point A yields.

How much of the current I from the emf flows through each resistor? The current divides such that the potential difference VA − VB must be the same along each of the three paths—and it must equal the emf ℰ. From the definition of resistance,

1 2 3 1 2 30 orI I I I I I I I +

ξ =I 1R1=I 2R2=I 3R3



Therefore, the currents are

ξ =I 1R1=I 2R2=I 3R3

I 1=ξR1

, I 2=ξR2

, I 3=ξR3

,

I= ξR1

+ ξR2

+ ξR3

,

Iξ = 1

R1

+ 1R2

+ 1R3

,



The three parallel resistors can be replaced by a single equivalent resistor Req. In order for the same current to flow, Req must be chosen so that ℰ = IReq. Then I/ℰ = 1/Req and

1REQ

= 1R1

+ 1R2

+ 1R3

,



For N resistors connected in parallel,

Note that the equivalent resistance for two or more resistors in parallel is smaller than any of the resistances (1/Req > 1/Ri, so Req < Ri ).

Note also that the equivalent resistance for resistors in parallel is found in the same way as the equivalent capacitance for capacitors in series.

eq 1 2

1 1 1 1 1

i NR R R R R




Fro example, with two resistors:

You can show that:

!! In the same way, making the same argument,

convince me that:

1Req

= 1R1

+ 1R2

Req=R1R2

R1+R2

=R1

R2

R1+R2

<R1

Req<R2


Example 18.6

(a) Find the equivalent resistance for the two resistors in the figure if R1 = 20.0 Ω and R2 = 40.0 Ω.

(b) What is the ratio of the current through R1 to the current through R2?



Points A and B are at the same potential; points C and D are at the same potential. Therefore, the voltage drops across the two resistors are equal; the two resistors are in parallel.

The ratio of the currents can be found by equating the potential differences in the two branches in terms of the current and resistance.




1

eq 1 2

eq 1

1 1 1 1 1a 0 0750

20 0 40 0

113 3

0 0750

R R R

R

1 1 2 2

1 2

2 1

b

40 02 00

20 0

I R I R

I R

I R


Example 18.7

(a) Find the equivalent resistance for the network of resistors in the figure.

(b) Find the current through the resistor R2 if ℰ = 0.60 V.



Example 18.7 Strategy 1

Simplify the network of resistors in a series of steps.

At first, the only series or parallel combination is the two resistors (R3 and R4) in parallel between points B and C.

No other pair of resistors has either the same current (for series) or the same voltage drop (for parallel).



We replace those two with an equivalent resistor, redraw the circuit, and look for new series or parallel combinations, continuing until the entire network reduces to a single resistor.



(a)

1 1

eq3 4

1 1 1 12 0

6 0 3 0R

R R



(a) continued

eq 4 0 2 0 6 0R



(a) continued

1

eq

1 13 6

6 0 9 0R



(b) The current through R2 is I2.



(b) continued. When I2 flows through an equivalent resistance of 6.0 Ω, the voltage drop is 0.60 V.

2

0 60 V0 10 A

6 0I


Emfs in Parallel 1

Two or more sources of equal emf are often connected in parallel with all the positive terminals connected together and all the negative terminals connected together.

The equivalent emf for any number of equal sources in parallel is the same as the emf of each source.

The advantage of connecting sources in this way is not to achieve a larger emf, but rather to lower the internal resistance and thus supply more current.


Emfs in Parallel 2

Never connect unequal emfs in parallel or connect emfs in parallel with opposite polarities.

In such cases the two batteries quickly drain one another and supply little or no current to the rest of the circuit.


Capacitors in Parallel 1

Capacitors in series have the same charge but may have different potential differences.

Capacitors in parallel share a common potential difference but may have different charges.



1 2 3

1 2 3 1 2 3 1 2 3

eq

eq 1 2 3

Q q q q

Q q q q C C C C C C

C

q C V

V

Q C

C C C

%

% % % %

%

ΔV=ξQ=q1+q2+q3=C1ξ +C2ξ +C3ξ =(C1+C2+C3)ξ

Q=Ceqξ



For N capacitors connected in parallel,

!! Is it possible to argue that capacitors in series or parallel are larger or smaller than any individual capacitor (the way we did for resistors)?

eq 1 2i NC C C C C


18.7 Circuit Analysis Using Kirchhoff’s Rules

Sometimes a circuit cannot be simplified by replacing parallel and series combinations alone.

In such cases, we apply Kirchhoff’s rules directly and solve the resulting equations simultaneously.


Problem-Solving Strategy: Using Kirchhoff’s Rules to Analyze a Circuit 1

1. Replace any series or parallel combinations with their equivalents.

2. Assign variables to the currents in each branch of the circuit ( I1 , I2 , . . .) and choose directions for each current. Draw the circuit with the current directions indicated by arrows. It does not matter whether or not you choose the correct direction.

3. Apply Kirchhoff’s junction rule to all but one of the junctions in the circuit. (Applying it to every junction produces one redundant equation.) Remember that current into a junction is positive; current out of a junction is negative.



4. Apply Kirchhoff’s loop rule to enough loops so that, together with the junction equations, you have the same number of equations as unknown quantities.

For each loop, choose a starting point and a direction to go around the loop. Be careful with signs. For a resistor, if your path through a resistor goes with the current (“downstream”), there is a potential drop; if your path goes against the current (“upstream”), the potential rises.



For an emf, the potential drops or rises depending on whether you move from the positive terminal to the negative or vice versa; the direction of the current is irrelevant.

A helpful method is to write “ + ” and “ − ” signs on the ends of each resistor and emf to indicate which end is at the higher potential and which is at the lower potential.



5. Solve the loop and junction equations simultaneously. If a current comes out negative, the direction of the current is opposite to the direction you chose.

6. Check your result using one or more loops or junctions. A good choice is a loop that you did not use in the solution.


Example 18.8

Find the currents through each branch of the circuit in the figure.




First we look for series and parallel combinations.

R1 and ℰ1 are in series, but since one is a resistor and one an emf we cannot replace them with a single equivalent circuit element.



No pair of resistors is either in series or in parallel. R1 and R2 might look like they’re in parallel, but the emf ℰ1 keeps points A and F at different potentials, so they are not.



The two emfs might look like they’re in series, but the junction at point F means that the current through the two is not the same. Since there are no series or parallel combinations to simplify, we proceed to apply Kirchhoff’s rules directly.



Arbitrarily choose current directions through the resistors as shown: I1 up, I2 down, and I3 up.



1 3 2

1 2

2 3

0 1

4 0 6 0 1 5 V 0 2

6 0 3 0 3 0 V 0 3

I I I

I I

I I

1 3 2I I I

3 2 24 0 6 0 1 5 V 0I I I

3 24 0 10 0 1 5 V/ 1 5 AI I

3 212 0 30 0 4 5 A 3 Eq 2a I I

3 212 0 24 0 12 0 A 4 Eq 3 I I



Subtracting one from the other,

3 212 0 30 0 4 5 A 3 Eq 2a I I

3 212 0 24 0 12 0 A 4 Eq 3 I I

254 0 7 5 AI

2

7 5A 0 139 A

54 0I

34 10 0 139 A 1 5 AI

3

1 5 1 39A 0 723 A

4I



Equation (1) now gives I1:

1 3 2 0 723 A 0 139 A 0 584 AI I I



Rounded to two significant figures, the currents are I1 = +0.58 A, I3 = −0.72 A, and I2 = −0.14 A. Since I3 and I2 came out negative, the actual directions of the currents in those branches are opposite to the ones we arbitrarily chose.


18.8 Power and Energy in Circuits

From the definition of electric potential, if a charge q moves through a potential difference ΔV, the change in electric potential energy is

From energy conservation, a change in electric potential energy means that conversion between two forms of energy takes place.

For example, a battery converts stored chemical energy into electric potential energy. A resistor converts electric potential energy into internal energy.

EU q V


Power

The rate at which the energy conversion takes place is the power P.

Since current is the rate of flow of charge, I = q /Δt and

EU qV I VP

t t


Power-Supplied by an Emf

According to the definition of emf, if the amount of charge pumped by an ideal source of constant emf ℰ is q, then the work done by the battery is.

The power supplied by the emf is the rate at which it does work:

W=ξ q

P=ΔWΔ t

=ξ qΔ t

=ξ I


Energy Dissipated in a Resistor

From the definition of resistance, the potential drop across a resistor is

V = IR

Then the rate at which energy is dissipated (converted from an organized form to a disorganized form) in a resistor can be written

or

2 P I IR I R

2V VP V

R R


Power Supplied by an Emf with Internal Resistance

If the source has internal resistance, then the net power supplied is less than ℰI.

Some of the energy supplied by the emf is dissipated by the internal resistance. The net power supplied to the rest of the circuit is

P=ξ I−I 2 r=(ξ −Ir) I


Example 18.9 (1)

A flashlight is powered by two batteries in series. Each has an emf of 1.50 V and an internal resistance of 0.10 Ω. The batteries are connected to the light bulb by wires of total resistance 0.40 Ω. At normal operating temperature, the resistance of the filament is 9.70 Ω.

(a) Calculate the power dissipated by the bulb—that is, the rate at which energy in the form of heat and light flows away from it.

(b) Calculate the power dissipated by the wires and the net power supplied by the batteries.


Example 18.9 (2)

(c) A second flashlight uses four such batteries in series and the same resistance wires. A bulb of resistance 42.1 Ω (at operating temperature) dissipates approximately the same power as the bulb in the first flashlight.

Verify that the power dissipated is nearly the same and calculate the power dissipated by the wires and the net power supplied by the batteries.



All the circuit elements are in series. We can simplify the circuit by replacing all the resistors (including the internal resistances of the batteries) with one series equivalent and the two emfs with one equivalent emf. Doing so enables us to find the current. Then we can use

to find the power in the wires and in the filament.

The equation

gives the net power supplied by the batteries.

2P I IR I R




(a)

eq 3.00 V%

eq 9 70 0 40 2 0 10 10 30R

eq

eq

3 00 V0 2913 A

10 30I

R

%I=

ξ eq

Req

ξ eq=3 .00V



(a) continued. The power dissipated by the filament is.

(b)

22f 0 2913 A 9 70 0 823 WP I R

22w 0 2913 A 0 40 0 034 WP I R

2b eq eqP I I r %

0.20 eqr

2b 3 00 V 0 2913 A 0 2913 A 0 20 0 857 WP


(NOTE: If the power from the battery did not equal the dissipated power, we would know something was up.)



(c) and

The power dissipated by the filament is

which is only 0.1% more than the filament in the first flashlight.

eqIn the second circuit, 6.00 V%

eq 42 1 0 40 4 0 10 42 90R

eq

eq

6 00 V0 13986 A

42 90I

R

%

22f 0 13986 A 42 1 0 824 WP I R

I=ξ eq

Req

ξ eq=6.00V



(c) continued. The power dissipated by the wires is.

0.40 eqr

2b eq eq

6 00 V 0 13986 A 0 0078 W 0 831 W

P I I r

%Pb=ξ eq I−I2req

!! So what’s the point of using four batteries?

Pw=I2 req=(0.1396 A)2 x 0.40Ω=0.0078W


18.10 RC Circuits

Circuits containing both resistors and capacitors have many important applications.

RC circuits are commonly used to control timing. When windshield wipers are set to operate intermittently, the charging of a capacitor to a certain voltage is the trigger that turns them on. The time delay between wipes is determined by the resistance and capacitance in the circuit; adjusting a variable resistor changes the length of the time delay.

We can also use the RC circuit as a simplified model of the transmission of nerve impulses.


Charging RC Circuit 1

Switch S is initially open and the capacitor is uncharged.

When the switch is closed, current begins to flow and charge starts to build up on the plates of the capacitor.

At any instant, Kirchhoff’s loop law requires that

R C 0 V V%ξ −V R−V C=0



Just after the switch is closed, the potential difference across the resistor is equal to the emf since the capacitor is uncharged. Initially, a relatively large current I0 = ℰ/R flows.

As the voltage drop across the capacitor increases, the voltage drop across the resistor decreases, and thus the current decreases.

+F, vD



Long after the switch is closed, the potential difference across the capacitor is nearly equal to the emf and the current is small.

+FBATTFCAP

+

-



Using calculus, it can be shown that the voltage across the capacitor involves an exponential function:

where e 2.718 is the base of the natural logarithm and the quantity τ = RC is called the time constant for the RC circuit.

!! Show that RC has units of time.

V C (t )=ξ (1−e−t /τ )



We can use the voltage loop rule to find the current

Solving for I,

ξ −I (t )R−ξ (1−e−t /τ )=0

ξ −V R(t )−V C (t )=0

I (t )=ξRe−t /τ



Power

For a charging capacitor, the power P = IVC is the rate at which energy is being stored in the capacitor.

While a capacitor is charging, the emf supplies energy at a rate of P = Iℰ; this is equal to the sum of the rate that energy is dissipated in the resistor ( IVR ) and the rate that energy is stored in the capacitor ( IVC ), as expected because energy must be conserved.

Recall from CH 17 that only half the work done by the battery is stored on the capacitor – here we see that some of that energy is dissipated in a resistor (even if it’s only the resistance of the wires).


Example 18.10

Two 0.500-μF capacitors in series are connected to a 50.0-V battery through a 4.00-MΩ resistor at t = 0. The capacitors are initially uncharged.

(a) Find the charge on the capacitors at t = 1.00 s and t = 3.00 s.

(b) Find the current in the circuit at the same two times.



First we find the equivalent capacitance of two 0.500-μF capacitors in series. Then we can find the time constant using the equivalent capacitance. The equation

gives the voltage across the equivalent capacitor at any time t; once we know the voltage, we can find the charge from Q = CVC.

The charge on each of the two capacitors is equal to the charge on the equivalent capacitor. The current decreases exponentially according to

C 1 t /V t e %

t /I t eR

%

V C (t )=ξ (1−e−t /τ )

I (t )=ξRe−t /τ



(a)

eq

1 1 1 2

C C C C

eq

10 250 F

2C C

6 6eq 4 00 10 0 250 10 F 1 00 sRC

6 6f eq 0 250 10 F 50 0 V 12 5 10 C

12 5 C

Q C

%

eq C eq f1 1t / t /Q t C V t C e Q e %

Q f=Ceqξ =0.250⋅106F x50.0V=12.5⋅10−6C

Q (t )=CeqV C (t )=Ceqξ (1−e−t /τ )=Q f (1−e−t /τ )



(a) continued.

1.00 1.00f

3.00 3.00f

A

C

t 1.00 s, / 1.00; the charge on each capacitor is

At 3.00 s, / 3.00; the charge on each capacito

.1 12 5 μC 1 7.90 μC

1 12.5 1

r

.

1

i

μ 1 μ

s

C 9

Q

t

Q

t

e e

Q Q e e

t t



(b)0 6

50 0 V12 5 A

4 00 10I

R

%

0t /I I e

1 00 1 000

At 1

12 5 A 4 60

.00 s,

A

I I e e

t

3 00 3 000 1

At 3

2 5 A 0 62

0 s

2

A

. 0 ,

t

I I e e

I 0=ξR


Discharging RC Circuit 1

In the figure, the capacitor is first charged to a voltage ℰ by closing switch S1 with switch S2 open. Once the capacitor is fully charged, S1 is opened and then S2 is closed at t = 0.

Now the capacitor acts like a battery in the sense that it supplies energy in the circuit, though not at a constant potential difference. As the potential difference between the plates causes current to flow, the capacitor discharges.




The loop rule requires that the voltages across the capacitor and resistor be equal in magnitude.

As the capacitor discharges, the voltage across it decreases.

A decreasing voltage across the resistor means that the current must be decreasing. The current as a function of time is the same as for the charging circuit with time constant = RC. The voltage across the capacitor begins at its maximum value ℰ and decreases exponentially.



V C (t )=ξ e−t /τ



The current as a function of time is the same as in the charging circuit.

I (t )=ξRe−t /τ



Power

For a discharging capacitor, the energy stored in the capacitor decreases at a rate IVC and energy is dissipated in the resistor at an equal rate IVR = IVC, as expected from energy conservation.



©2020 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.


Chapter 18: Electric Current and Circuits

18.1 Electric Current.

18.2 Emf and Circuits.

18.3 Microscopic View of Current in a Metal: The Free-Electron Model.

18.4 Resistance and Resistivity.

18.5 Kirchhoff’s Rules.

18.6 Series and Parallel Circuits.

18.7 Circuit Analysis Using Kirchhoff’s Rules.

18.8 Power and Energy in Circuits.

18.9 Measuring Currents and Voltages.

18.10 RC Circuits.

18.11 Electrical Safety.





Recall this from CHAP 16:

Neutral

Charge moves(not really)

Thing 1 Thing 2

- +





What’s really going on...In this picture, five electrons (so a charge of ) move past surface A TO THE LEFT. So this is the effect of a positive current to the RIGHT.

8⋅10−19C


Definition of Current

The SI unit of current, equal to one coulomb per second, is the ampere (A).

qI

t

1 C 1 A s


Example 18.1

Two wires of cross-sectional area 1.6 mm2 connect the terminals of a battery to the circuitry in a clock. During a time interval of 0.040 s, 5.0 × 1014 electrons move to the right through a cross section of one of the wires. (Actually, electrons pass through the cross section in both directions; the number that cross to the right is 5.0 × 1014 more than the number that cross to the left.)

What is the magnitude and direction of the current in the wire?



Current is the rate of flow of charge.

We are given the number N of electrons; multiplying by the elementary charge e gives the magnitude of moving charge Δq.



14 19 5

5

5.0 10 1.60 10 C 8.0 10 C

8.0 10 C 0.0020 A 2.0 mA

0.040 s

q Ne

qI

t


Electric Current in Liquids and Gases

Electric currents can exist in liquids and gases as well as in solid conductors.

In an ionic solution, both positive and negative charges contribute to the current by moving in opposite directions.

Since positive and negative charges are moving in opposite directions, they both contribute to current in the same direction.


Application : Current in Neon Signs and Fluorescent Lights


18.2 Emf and Circuits

To maintain a current in a conducting wire, we need to maintain a potential difference between the ends of the wire.

One way to do that is to connect the ends of the wire to the terminals of a battery (one end to each of the two terminals).

An ideal battery maintains a constant potential difference between its terminals, regardless of how fast it must pump charge to do so. An ideal battery is analogous to an ideal water pump that maintains a constant pressure difference between intake and output regardless of the volume flow rate.


Circuit Symbols for a Battery

Of the two vertical lines, the long line represents the terminal at higher potential and the short line represents the terminal at lower potential.


Electromotive Force

The potential difference maintained by an ideal battery is called the battery’s emf (symbol ℰ).

Emf originally stood for electromotive force , but emf is not a measure of the force applied to a charge or to a collection of charges; emf cannot be expressed in newtons.

Rather, emf is measured in units of potential (volts) and is a measure of the work done by the battery per unit charge. To avoid this confusion, we just write “emf” (pronounced ee-em-ef).


Work Done by an Ideal Battery

If the amount of charge pumped by an ideal battery of emf ℰ is q, then the work done by the battery is.

Work done by an ideal battery:

W=ξ q



MinnesotaGulf

MississippiRiver


Batteries

Batteries come with various emfs (12 V, 9 V, 1.5 V, etc.) as well as in various sizes. The size of a battery does not determine its emf. (Notice that most batteries are a multiple of 1.5 V)

Common battery sizes AAA, AA, A, C, and D all provide the same emf (1.5 V). However, the larger batteries have a larger quantity of the chemicals and thus store more chemical energy.





According to the company, the Samsung S21 Ultracell phone has a battery “size” of 5000 mAh. What is this? Check the units:

mAh = milli-Amp-hour =

So this “size” is not a physical size, but how much charge the battery can “lift” to the higher voltage. (But it DOES correlate with the physical size).

(5 A)(3600 s)=18⋅103C



So this “size” is not a physical size, but how much charge the battery can “lift” to the higher voltage.

Cell phone batteries usually have an “emf” of 3.7 V (note this breaks my “multiple of 1.5 V” rule – cell phone batteries use a different chemistry and so have a different emf)

So the S21 battery can raise 18,000 C of charge to a potential 3.7 V higher than the “ground” of the battery. So the total work the battery can do is:

(about the same as 2 mL = 2 milliliters of gasoline, which is about the volume of three nickels)

W=ξ q=(3.7V )(18⋅103C)=67 kJ



So the total work the battery can do is:

According to Tom’s Guide, the S21 Ultra battery lasts about 10 hours (= 36,000 sec), so the power provided by the battery is:

W=ξ q=(3.7V )(18⋅103C)=67 kJ

P= energytime

= 67 kJ36 ksec

=1.9W


Circuits




Circuits


Current is a flow of charge, and as far as we know, charge is conserved (never created or destroyed).

Energy is conserved – an electrical circuit is generally a way to convert electrical energy into some other form:Light (from a bulb), Motion (an electric fan), Heat (an electric stove)

Electrical energy can come from:Light (a photovoltaic), Motion (a windmill or dam), Heat (a solar thermal or thermoelectric device)Civilization is about converting energy from one form to another.



Direct Current

In this chapter, we consider only circuits in which the current in any branch always moves in the same direction—a direct current (dc) circuit.

In Chapter 21, we study alternating current (ac) circuits, in which the currents periodically reverse direction.


18.3 Microscopic View of Current in a Metal: The Free-Electron Model

The electrons have a nonzero average velocity called the drift velocity, vD. The magnitude of the drift velocity (the drift speed ) is much smaller than the instantaneous speeds of the electrons.


The Drift Velocity is Constant

An electron has a uniform acceleration between collisions, but every collision sends it off in some new direction with a different speed. Each collision between an electron and an ion is an opportunity for the electron to transfer some of its kinetic energy to the ion.

The net result is that the drift velocity is constant, and energy is transferred from the electrons to the ions at a constant rate.



The number of electrons in the volume is N = nAvDΔt ; the magnitude of the charge is.

DQ Ne neA t



Therefore, the magnitude of the current in the wire is.

Remember that, since electrons carry negative charge, the direction of current flow is opposite the direction of motion of the electrons. The electric force on the electrons is opposite the electric field, so the current is in the direction of the electric field in the wire.

DQ Ne neA t

D

QI neA

t



The equation

can be generalized to systems in which the current carriers are not necessarily electrons, simply by replacing e with the charge of the carriers:

The quantities v+ and v− are drift speeds —both are positive.

D

QI neA

t

I n eA n eA


Example 18.2

A #12 gauge copper wire, commonly used in household wiring, has a diameter of 2.053 mm. There are 8.00 × 1028 conduction electrons per cubic meter in copper.

If the wire carries a constant dc current of 5.00 A, what is the drift speed of the electrons?



2 21

4A r d

228 3 19 3

4 1

5.00 A1

8.00 10 m 1.602 10 C 2.053 10 m4

1.179 10 m s 0.118 mm/s

D

I

neA

SNAIL: ~3-13 mm/s TORTOISE: ~ 110 mm/s


18.4 Resistance and Resistivity

Suppose we maintain a potential difference across the ends of a conductor. How does the current I that flows through the conductor depend on the potential difference ΔV across the conductor?

For many conductors, the I is proportional to ΔV.



Ohm’s law is not a universal law of physics like the conservation laws. THEN WHY CALL IT THAT!

It does not apply at all to some materials, whereas even materials that obey Ohm’s law equation for a wide range of potential differences fail to do so when Δ V becomes too large.

Any homogeneous material follows Ohm’s law equation for some range of potential differences; metals that are good conductors follow Ohm’s law equation over a wide range of potential differences.

I V



TL; DR

Ohm’s law equation works when it works and not when it doesn’t – it will ALWAYS work over a limited range (big or small) and NEVER works for all conditions.

I V


Resistance

The electrical resistance R is defined to be the ratio of the potential difference (or voltage ) ΔV across a conductor to the current I through the material.

Definition of resistance (NOT Ohm’s law equation):

In SI units, electrical resistance is measured in ohms (symbol Ω, the Greek capital omega), defined as.

VR

I

1 1 V/A


Resistance

Think of “resistance” like a drag force for moving charges – in the same way that water “resists” you walking way more than air “resists” you walking, some materials “resist” the passage of charged particles more or less – Essentially, it is the number of things that the electron (or whatever) collides with that prevent it from moving to lower potential energy in the way that air drag slows down a falling object as the falling object collides with air molecules.


Ohmic and Nonohmic Conductors

An ohmic conductor—one that follows Ohm’s law equation—has a resistance that is constant, regardless of the potential difference applied.

For an ohmic conductor, a graph of current versus potential difference is a straight line through the origin with slope 1/R.

For some nonohmic systems, the graph of I versus ΔV may be dramatically nonlinear.


Ohmic Conductors: Example


Nonohmic Conductors: Example



Resistivity 1

Resistance depends on size and shape. We expect a long wire to have higher resistance than a short one (in a longer wire, there are more things to bang into – imagine a long road, which has more stop signs to slow you down.) and a thicker wire to have a lower resistance than a thin one (like a multi-lane road which has more paths to follow to lower potential energy).

The electrical resistance of a conductor of length L and cross-sectional area A can be written:

Resistance = [properties of material] x [geometry of resistor]

LR

A


Resistivity 2

The constant of proportionality ρ (Greek letter rho), which is an intrinsic characteristic of a particular material at a particular temperature, is called the resistivity of the material.

The SI unit for resistivity is Ω·m. (Be prepared to argue why that MUST be true)

The inverse of resistivity is called conductivity [SI units (Ω·m) −1].

LR

A


Resistivities for Selected Materials

Table 18.1 Resistivities and Temperature Coefficients at 20°C.

ρ (Ω·m) α (°C–1) ρ (Ω·m) α (°C–1)

Conductors Semiconductors (pure)

Silver 1.59 × 10−8 3.8 × 10−3 Carbon 3.5 × 10−5 −0.5 × 10–3

Copper 1.67 × 10−8 4.05 × 10−3 Germanium 0.6 −50 × 10–3

Gold 2.35 × 10−8 3.4 × 10−3 Silicon 2300 −70 × 10–3

Aluminum 2.65 × 10−8 3.9 × 10−3

Tungsten 5.40 × 10−8 4.50 × 10−3

Iron 9.71 × 10−8 5.0 × 10−3 Insulators

Platinum 10.6 × 10−8 3.64 × 10−3 Wood 108 − 1011

Lead 21 × 10−8 3.9 × 10−3 Glass 1010 − 1014

Manganin 44 × 10−8 0.002 × 10−3 Rubber (hard) 1013 − 1016

Constantan 49 × 10−8 0.002 × 10−3 Lucite > 1013

Mercury 96 × 10−8 0.89 × 10−3 Teflon > 1013

Nichrome 108 × 10−8 0.4 × 10−3 Quartz (fused) > 1016


Resistivity of Water

The resistivity of water depends strongly on the concentration of ions.

Pure water contains only the ions produced by self-ionization (H2O ↔H+ + OH− ). As a result, pure water is an insulator. (ONLY BECAUSE there is such a small number of ions)

Even a small amount of dissolved minerals dramatically lowers the resistivity. The resistivity is so sensitive to the concentration of impurities that resistivity measurements are used to determine water purity.


Example 18.3

(a) A 30.0-m-long extension cord is made from two #19-gauge copper wires. (The wires carry currents of equal magnitude in opposite directions.) What is the resistance of each wire at 20.0°C? The diameter of #19-gauge wire is 0.912 mm.

(b) If the copper wire is to be replaced by an aluminum wire of the same length, what is the minimum diameter so that the new wire has a resistance no greater than the old?



(a)

8

22 4 7 2

1.67 10 m

1 19.12 10 m 6.533 10 m

4 4A d

8

7 2

1.67 10 m 30.0 m

6.533 10 m0.767

LR

A



(b)a c

a c

2 2a c

2 2a c c a

1 14 4

R R

L L

d d

d d

8a

a c 8c

2.65 10 m0.912 mm 1.15 mm

1.67 10 md d


Resistors 1

A resistor is a circuit element designed to have a known resistance.

In circuit analysis, it is customary to write the relationship between voltage and current for a resistor as V = IR. (which is good, since that’s the definition…..)

Remember that V actually stands for the potential difference between the ends of the resistor even though the symbol Δ is omitted.

©Artit Thongchuea/Shutterstock


Resistors 2

Current in a resistor flows in the direction of the electric field, which points from higher to lower potential.

Therefore, if you move across a resistor in the direction of current flow, the voltage drops by an amount IR.

In a circuit diagram, the symbol.

represents a resistor or any other device in a circuit that dissipates electric energy.

A straight line in a circuit diagram represents a conducting wire with negligible resistance.



When the current through a source of emf is zero, the terminal voltage—the potential difference between its terminals—is equal to the emf.

When the source supplies current to a load (a light bulb, a toaster, or any other device that uses electric energy), its terminal voltage is less than the emf; there is a voltage drop due to the internal resistance of the source.

If the current is I and the internal resistance is r, then the voltage drop across the internal resistance is Ir and the terminal voltage is

V B=ξ −I r



Two rules, developed by Gustav Kirchhoff (1824 to 1887), are essential in circuit analysis.

Kirchhoff’s junction rule states that the sum of the currents that flow into a junction - any electric connection - must equal the sum of the currents that flow out of the same junction.

The junction rule is a consequence of the law equation of conservation of charge. Since charge does not continually build up at a junction, the net rate of flow of charge into the junction must be zero.


Kirchhoff’s Junction Rule

outin 0I I



Kirchhoff’s loop rule is an expression of energy conservation applied to changes in potential in a circuit.

Recall that the electric potential must have a unique value at any point; the potential at a point cannot depend on the path one takes to arrive at that point.

Therefore, if a closed path is followed in a circuit, beginning and ending at the same point, the algebraic sum of the potential changes must be zero.


Kirchhoff’s Loop Rule

For any path in a circuit that starts and ends at the same point. (Potential rises are positive; potential drops are negative.)

If you follow a path through a resistor going in the same direction as the current, the potential drops (ΔV = − IR). If your path takes you through a resistor in a direction opposite to the current (“upstream”), the potential rises (ΔV = + IR). For an emf, the potential drops if you move from the positive terminal to the negative (ΔV = −ℰ); it rises if you move from the negative to the positive (ΔV = +ℰ).

0V


Kirchhoff’s “Rules”

Loop “rule” says that a charge can gain or lose electrical energy in a circuit, but that all GAINS are balanced by LOSSES – conserved!(Technically, this is electrical energy gained and lost – it is really only converted – a battery converts chemical energy to electrical energy and a resistor converts electrical energy into sound/light/heat/whatever)

Junction “rule” says that you can’t create or destroy charge – conserved!

These “rules” come from conservation laws (and Ohm’s “law” is just a rule of thumb that works when it works...)


18.6 Series and Parallel Circuits

When one or more electric devices are wired so that the same current flows through each one, the devices are said to be wired in series.



The circuit shows two resistors in series. The straight lines represent wires, which we assume to have negligible resistance.

Negligible resistance means negligible voltage drop (V = IR), so points connected by wires of negligible resistance are at the same potential.

The junction rule, applied to any of the points A – D, tells us that the same current flows through the emf and the two resistors.



Let’s apply the loop rule to a clockwise loop DABCD . From D to A we move from the negative terminal to the positive terminal of the emf, so ΔV = +1.5 V. Since we move around the loop with the current, the potential drops as we move across each resistor.

1 21 5 V 0IR IR

1 2 1 5 VI R R



1 2Req R R

eq 1 2 1 5 VIR I R R



For any number N of resistors connected in series,

Note that the equivalent resistance for two or more resistors in series is larger than any of the resistances.

eq i 1 2 NR R R R R


Emfs in Series 1

In many devices, batteries are connected in series with the positive terminal of one connected to the negative terminal of the next.

This provides a larger emf than a single battery can.


Emfs in Series 2

The emfs of batteries connected in this way are added just as series resistances are added. However, there is a disadvantage in connecting batteries in series: the internal resistance is larger because the internal resistances are in series as well.


Emfs in Series 3

Sources can be connected in series with the emfs in opposition. A common use for such a circuit is in a battery charger. In the figure, as we move from point C to B to A, the potential decreases by ℰ2 and then increases by ℰ1, so the net emf is ℰ1 − ℰ2.




The figure shows two capacitors connected in series. Although no charges can move through the dielectric of a capacitor from one plate to the other, the instantaneous currents I that flow onto one plate and from the other must be equal.

Why? The two plates of a capacitor always have charges of equal magnitudes and opposite signs. Therefore, the magnitudes of the charges on the two plates must change at the same rate . The rate of change of the charge is equal to the current. Viewed from the outside, the capacitor behaves as if a current I flows through it.



We want to find the equivalent capacitance Ceq that would store the same amount of charge as each of the series capacitors for the same applied voltage.

With the switch closed, the emf pumps charge so that the potential difference between points A and B is equal to the emf. The capacitors are fully charged and the current goes to zero. From Kirchhoff’s loop rule,


ξ −V 1−V 2=0



The equivalent capacitance is defined by ℰ = Q/ Ceq.

1 21 2

andQ Q

V VC C

eq 1 2

eq 1 2

0

1 1 1

Q Q Q

C C C

C C C

ξ −V 1−V 2=0



For N capacitors connected in series,

Note that the equivalent capacitor stores the same magnitude of charge as each of the capacitors it replaces.

eq 1 2

1 1 1 1 1

i NC C C C C



When one or more electrical devices are wired so that the potential difference across them is the same, the devices are said to be wired in parallel.



In the figure, an emf is connected to three resistors in parallel with each other. The left side of each resistor is at the same potential since they are all connected by wires of negligible resistance.

Likewise, the right side of each resistor is at the same potential. Thus, there is a common potential difference across the three resistors.




Applying the junction rule to point A yields.

How much of the current I from the emf flows through each resistor? The current divides such that the potential difference VA − VB must be the same along each of the three paths—and it must equal the emf ℰ. From the definition of resistance,

1 2 3 1 2 30 orI I I I I I I I +

ξ =I 1R1=I 2R2=I 3R3



Therefore, the currents are

ξ =I1R1=I 2R2=I 3R3

I 1=ξR1

, I 2=ξR2

, I 3=ξR3

,

I= ξR1

+ ξR2

+ ξR3

,

Iξ = 1

R1

+ 1R2

+ 1R3

,



The three parallel resistors can be replaced by a single equivalent resistor Req. In order for the same current to flow, Req must be chosen so that ℰ = IReq. Then I/ℰ = 1/Req and

1REQ

= 1R1

+ 1R2

+ 1R3

,



For N resistors connected in parallel,


Note also that the equivalent resistance for resistors in parallel is found in the same way as the equivalent capacitance for capacitors in series.

eq 1 2

1 1 1 1 1

i NR R R R R




Fro example, with two resistors:

You can show that:

!! In the same way, making the same argument,

convince me that:

1Req

= 1R1

+ 1R2

Req=R1R2

R1+R2

=R1

R2

R1+R2

<R1

Req<R2


Example 18.6

(a) Find the equivalent resistance for the two resistors in the figure if R1 = 20.0 Ω and R2 = 40.0 Ω.

(b) What is the ratio of the current through R1 to the current through R2?



Points A and B are at the same potential; points C and D are at the same potential. Therefore, the voltage drops across the two resistors are equal; the two resistors are in parallel.

The ratio of the currents can be found by equating the potential differences in the two branches in terms of the current and resistance.




1

eq 1 2

eq 1

1 1 1 1 1a 0 0750

20 0 40 0

113 3

0 0750

R R R

R

1 1 2 2

1 2

2 1

b

40 02 00

20 0

I R I R

I R

I R


Example 18.7

(a) Find the equivalent resistance for the network of resistors in the figure.

(b) Find the current through the resistor R2 if ℰ = 0.60 V.




Simplify the network of resistors in a series of steps.

At first, the only series or parallel combination is the two resistors (R3 and R4) in parallel between points B and C.

No other pair of resistors has either the same current (for series) or the same voltage drop (for parallel).



We replace those two with an equivalent resistor, redraw the circuit, and look for new series or parallel combinations, continuing until the entire network reduces to a single resistor.



(a)

1 1

eq3 4

1 1 1 12 0

6 0 3 0R

R R



(a) continued

eq 4 0 2 0 6 0R



(a) continued

1

eq

1 13 6

6 0 9 0R



(b) The current through R2 is I2.



(b) continued. When I2 flows through an equivalent resistance of 6.0 Ω, the voltage drop is 0.60 V.

2

0 60 V0 10 A

6 0I


Emfs in Parallel 1

Two or more sources of equal emf are often connected in parallel with all the positive terminals connected together and all the negative terminals connected together.

The equivalent emf for any number of equal sources in parallel is the same as the emf of each source.

The advantage of connecting sources in this way is not to achieve a larger emf, but rather to lower the internal resistance and thus supply more current.


Emfs in Parallel 2

Never connect unequal emfs in parallel or connect emfs in parallel with opposite polarities.

In such cases the two batteries quickly drain one another and supply little or no current to the rest of the circuit.



Capacitors in series have the same charge but may have different potential differences.

Capacitors in parallel share a common potential difference but may have different charges.



1 2 3

1 2 3 1 2 3 1 2 3

eq

eq 1 2 3

Q q q q

Q q q q C C C C C C

C

q C V

V

Q C

C C C

%

% % % %

%

ΔV=ξQ=q1+q2+q3=C1ξ +C2ξ +C3ξ =(C1+C2+C3)ξ

Q=Ceqξ



For N capacitors connected in parallel,

!! Is it possible to argue that capacitors in series or parallel are larger or smaller than any individual capacitor (the way we did for resistors)?

eq 1 2i NC C C C C


18.7 Circuit Analysis Using Kirchhoff’s Rules

Sometimes a circuit cannot be simplified by replacing parallel and series combinations alone.

In such cases, we apply Kirchhoff’s rules directly and solve the resulting equations simultaneously.



1. Replace any series or parallel combinations with their equivalents.

2. Assign variables to the currents in each branch of the circuit ( I1 , I2 , . . .) and choose directions for each current. Draw the circuit with the current directions indicated by arrows. It does not matter whether or not you choose the correct direction.

3. Apply Kirchhoff’s junction rule to all but one of the junctions in the circuit. (Applying it to every junction produces one redundant equation.) Remember that current into a junction is positive; current out of a junction is negative.