college physics bcrede/files/fall2014/electric_currents... · the ﬁgure shows a family of...

CollegePhysics B

EquipotentialSurfaces

Capacitors

Capacitors in Series

Capacitors in Parallel

ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series

Resistors in Parallel

College Physics B - PHY2054C

Capacitors and Electric Currents

09/10/2014

My Office Hours:

Tuesday 10:00 - Noon

206 Keen Building

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


PHY2054C

First Mini-Exam next week on Wednesday!!

• Location: UPL 101

• Equation sheet will be provided.

• Bring a picture ID to the exam.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Example: Problem 18.1

If the electric field is zero in a particular region of space, what

does that tell you about the electric potential in that region? Is

the potential zero, constant, or something else? Explain.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series



If the electric field is zero in a particular region of space, what

does that tell you about the electric potential in that region? Is

the potential zero, constant, or something else? Explain.

E = −∆V

∆x

V =PE elec

q

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series



Will the electric field always be zero at any point where the

electric potential is zero? Why or why not?

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series



Will the electric field always be zero at any point where the

electric potential is zero? Why or why not?

E = −∆V

∆x

V =PE elec

q

Assume origin is in the middle

between the two charges:

V =k q

r+

−(k q)

r

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Outline

1 Equipotential Surfaces

2 Capacitors



3 Electric Currents

4 Ohm’s Law

Resistance

Electric Power

5 Electric Circuits

Resistors in Series


CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Equipotential Surfaces

A useful way to visualize electric fields is through plots of

equipotential surfaces:

• Contours where the electric potential is constant.

• Equipotential lines are in two-dimensions.

The equipotential surfaces are always perpendicular to the

direction of the electric field.

• For motion parallel to an equipotential surface, V is

constant and ∆V = 0.

• Electric field component parallel to the surface is zero.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Example: Point Charge

The electric field lines emanate

radially outward from the charge.

• The equipotential surfaces

are perpendicular to the field.

• The equipotentials are a

series of concentric spheres.

• Different spheres correspond

to different values of V .

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Example: Dipole

The dipole consists of charge

+q and −q.

• Field lines are plotted in

blue.

• Equipotential lines are

plotted in orange.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Question 1

The figure shows a family of equipotential surfaces.

If V1 > V2 > V3 > V4, is the object in the figure positively

charged or negatively charged?

A positive

B negative

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Question 1

The figure shows a family of equipotential surfaces.

If V1 > V2 > V3 > V4, is the object in the figure positively

charged or negatively charged?

A positive

B negative

• The electric field is directed from regions of high potential

to regions of low potential.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Question 2

An ion is released from rest and moves due to the force from

an electric field from a position in the field having a potential of

14 V to a position having a potential of 8 V. The ion:

A must have a positive charge.

B must have a negative charge.

C can have either a positive or a negative charge.

D must be neutral.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Outline


2 Capacitors



3 Electric Currents

4 Ohm’s Law

Resistance

Electric Power

5 Electric Circuits

Resistors in Series


CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Capacitors

A capacitor can be used to store

charge and energy.

1 Each plate produces a field:

E =Q

2ǫ0 A

2 In the region between the

plates, the fields from the

two plates add, giving:

E =Q

ǫ0 A=

∆V

d,

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Capacitors

A capacitor can be used to store

charge and energy.

1 Each plate produces a field:

E =Q

2ǫ0 A

2 In the region between the

plates, the fields from the

two plates add, giving:

E =Q

ǫ0 A=

∆V

d,

where d is the distance

between the plates.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Capacitance

Electric Potential of a capacitor:

E =Q

ǫ0 A=

∆V

d

∆V =Q d

ǫ0 A= E d

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Capacitance

Electric Potential of a capacitor:

E =Q

ǫ0 A=

∆V

d

∆V =Q d

ǫ0 A= E d

Capacitance C is defined as:

∆V =Q

C

C =ǫ0 A

d

”Parallel-Plate Capacitor”

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Capacitance

Capacitance C is defined as:

∆V =Q

C

C =ǫ0 A

d

• Units are the Farad or [F]:

1 F = 1 C / V.

(in honor of Michael Faraday)

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Capacitance

The total energy corresponds to area under ∆V − Q graph:

PE cap =1

2Q (∆V ) =

1

2C (∆V )2 =

1

2

Q 2

C

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series



When dealing with multiple capacitors, equivalent capacitance

is useful (V = Q/C):

∆V total = ∆V top + ∆V bottom

1

C equiv.

=1

C1+

1

C2

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series



When dealing with multiple capacitors, equivalent capacitance

is useful (V = Q/C):

Q total = Q1 + Q2

C equiv. = C1 + C2

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Outline


2 Capacitors



3 Electric Currents

4 Ohm’s Law

Resistance

Electric Power

5 Electric Circuits

Resistors in Series


CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Electric Current

The motion of charges leads to

the idea of electric circuits:

• Electric current, I, in a wire is

defined as the net amount of

charge that passes through it

per unit time at any point:

I =∆Q

∆t

• The unit of electric current:[

C

s

]

= [A ] Ampere

• Current is defined in terms of

net positive charge flow.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Electric Current

André-Marie Ampère

(22 January 1775 - 10 June 1836)

Electric current is the flow of electric charge (a phenomenon)

or the rate of flow of electric charge (a quantity). This flowing

electric charge is typically carried by moving electrons, in a

conductor such as wire; in an electrolyte, it is instead carried

by ions, and, in a plasma, by both.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Direction of the Current

+ -

current flow

device

[battery symbol]

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Direction of the Current

1 If the current is carried by positive charges moving with a

given velocity, the direction of the current is parallel to the

velocity.

2 If the current is carried by negative charges, the direction

of the current is opposite the charges’ velocity.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Current and Potential Energy

For charge to move along a wire, the electric potential energy

at one end of the wire must be higher than the electric potential

energy at the other end.

• Electric potential is related

to electric potential energy:

V = PEelec /q

• The potential is referred to

simply as “voltage”.

• The direction of I is always

from high to low potential,

regardless if the current is

carried by + or − charges.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Simple Circuit

If the battery terminals are connected to two ends of a wire, a

current is produced:

• Electrons move out of the negative terminal of the battery

through the wire and into the positive battery terminal.

• The chemical reaction moves charge internally between

the electrodes.

• No net charge accumulates on the battery terminals while

the current is present.

• Battery will “run down”.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Outline


2 Capacitors



3 Electric Currents

4 Ohm’s Law

Resistance

Electric Power

5 Electric Circuits

Resistors in Series


CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Ohm’s Law

V

I

ohmic device

slope = R

1 Drag force on electrons leads to a drift velocity proportional

to the force pushing the electrons.

2 Force is proportional to the electric field, so the drift velocity

is proportional to the field.

3 The electric field is proportional to the potential difference,

so the drift velocity is proportional to the potential difference.

4 The current is proportional to the drift velocity, so the current

is proportional to the potential difference:

I =V

ROhm′s Law

Unit of Resistance R :

[

V

A

]

= [Ω ]

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Ohm’s Law

George Simon Ohm

(16 March 1789 - 6 July 1854)

Ohm’s law states that the current through a conductor between

two points is directly proportional to the potential difference or

voltage across the two points, and inversely proportional to the

resistance between them.

Ohm’s Law:

http://phet.colorado.edu/sims/ohms-law/ohms-law_en.html

http://phet.colorado.edu/sims/ohms-law/ohms-law_en.html

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Resistivity

The resistivity, ρ, depends only on the material used to make

the wire. Resistance of a wire of length L and cross sectional

area A is given by:

R = ρL

AMaterial ρ [ Ω · m ]

Copper 1.7 × 10−8

Glass 1 to 1000 × 109

Silicon 0.1 to 100

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Resistors

All electronic devices, from heaters to light bulbs to stereo

amplifiers, offer resistance to the flow of current and are

therefore considered resistors.

• Resistors can be made in

many shapes and sizes.

• Each will have a resistance

proportional to the current

through and the potential

across the resistor.

Many, but not all, materials and devices obey Ohm’s Law.

Ohm’s Law is not a fundamental law of nature.

Resistors do obey Ohm’s Law.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Circuit Schematic

• The circuit diagram (A) shows

the symbols for the resistor

and the battery.

• Since the resistance of the

wires is much smaller than

that of the resistors, a good

approximation is Rwire = 0.

• If the circuit is open, there is

no current flow anywhere in

the circuit.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Circuit Symbols

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Electric Power

Reminder:

Ohm’s Law: R = V / I

Energy in a Resistor

• The test charge gained energy when it passed through the

battery.

• It lost energy as it passed through the resistor.

• Energy is converted into heat energy inside the resistor:

• The energy is dissipated as heat.

• It shows up as a temperature increase of the resistor and its

surroundings.

P (Power) =energy transformed

time=

Q V

t= I V

P = I V = I 2 R = V 2 /R

Electric Power

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Resistance of a Light Bulb

What is a typical household ligh bulb?

60 Watt light bulb

What is a typical household voltage?

110 Volts

What else do we know?

P = V I = V 2 /R

R =V 2

P=

(110 V)2

60 W≈ 200 Ω

Battery-Resistor Circuit:

phet.colorado.edu/en/simulation/battery-resistor-circuit

phet.colorado.edu/en/simulation/battery-resistor-circuit

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Resistivity & Temperature

Resistance of a metal wire: R = ρ LA

In general, the resistance of metal increases with temperature:

ρT = ρ0 [ 1 + α (T − T0) ]

Temperature Coefficients

Material α [ (C)−1 ]

Silver 0.0061

Copper 0.0068

Silicon −0.07

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Outline


2 Capacitors



3 Electric Currents

4 Ohm’s Law

Resistance

Electric Power

5 Electric Circuits

Resistors in Series


CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Resistors in Series

When current passes through one resistor and then another,

the resistors are said to be in series:

E − I R 1 − I R 2 = 0 Kirchhoff ′s Loop Rule

Any number of resistors can be connected in series. The

resistors will be equivalent to a single resistor with:

R equiv = R 1 + R 2 + R 3 + ...

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series



In some circuits, the current can take multiple paths:

• The different paths are called branches.

• The arrangement of resistors shown is called resistors in

parallel.

• Amount of current entering a junction must be equal to

the current leaving it.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series



Applying the Junction Rule (Kirchhoff ’s Junction Rule)

For path 1, +E − I 1 R 1 = 0

For path 2, +E − I 2 R 2 = 0

The total current is: I 3 = I 1 + I 2 = E

R 1+ E

R 2= E ( 1

R 1+ 1

R 2)

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Equivalent Resistance - Parallel

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Circuit Analysis

1 Some complex circuits can be solved by combinations of

series and parallel rules.

2 Other circuits must be analyzed directly by Kirchhoff’s Rules.

• Loop Rule: The total change in the electric potential around

any closed circuit path must be zero.• Junction Rule: The current entering a circuit junction must

equal the current leaving the junction.

3 Connecting resistors in series always gives a total resistance

larger than the resistance of any of the component resistors.

4 Connecting resistors in parallel always gives a total

resistance smaller than the resistance of any of the

component resistors.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Ammeters

An Ammeter is a device that

measures current.

• An ammeter must be connected in series with the

desired circuit branch.

• An ideal ammeter will measure current without changing

its value.

Must have a very low resistance.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Voltmeters

A Voltmeter is a device that

measures the voltage across

a circuit element.

• It must be connected in parallel with the element.

• An ideal voltmeter should measure the voltage without

changing its value.

Should have a very high resistance.

CollegePhysics B


Capacitors



ElectricCurrents

Ohm’s Law

Resistance

Electric Power

ElectricCircuits

Resistors in Series


Electric Currents and Nerves

Many nerves are long and thin, much like wires.

• The conducting solution inside the fiber acts as a resistor.

• The lipid layer acts as a capacitor.

• The nerve fiber behaves as an RC circuit.

More on RC circuits next week!

Your body is a moderately good conductor of electricity.

• The body’s resistance when dry is about 1500 Ω.

• When wet, the body’s resistance is about 500 Ω.

• Current is carried by different parts of the body:

Skin, internal organs, ...

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