electric current and dc circuits

Physics 201Professor P. Q. Hung

311B, Physics Building

Physics 201 – p. 1/34

Electric Current and DC Circuits

Summary of last lecture

Equipotential surfaces: Surfaces where thepotential is the same everywhere, e.g. thesurface of a conductor.

Q = C|∆V |. C: Capacitance, the capacity tostore charge

U = 12QV = 1

2CV 2 = Q2

2C : potential energy of acapacitor

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Beyond electrostaticA current of 0.2 mA coming from a 3.0 V batteryoperates a calculator for one hour. How muchcharge flows in the circuit?

In previous lectures, a conductor inelectrostatic equilibrium: No electric fieldinside ⇒ Conduction electrons do not flow.

For conduction electrons to start flowingtogether (current) in a given direction, weneed an electric field.

Difference in potential ⇒ Electric field.

Physics 201 – p. 3/34


Beyond electrostaticA current of 0.2 mA coming from a 3.0 V batteryoperates a calculator for one hour. How muchcharge flows in the circuit?

In previous lectures, a conductor inelectrostatic equilibrium: No electric fieldinside ⇒ Conduction electrons do not flow.

For conduction electrons to start flowingtogether (current) in a given direction, weneed an electric field.

Difference in potential ⇒ Electric field.Physics 201 – p. 3/34


Electric Current

Difference in potential ⇒ Electric field ⇒Conduction electrons move.

How do we create such a potentialdifference? By connecting the two ends of thewire to the two terminals of a battery whichposesses an electric potential difference.

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Electric Current

How does a battery create a potentialdifference between its two terminals? Bychemical reactions which transfer electronsfrom one terminal (making it positivelycharged) ( higher potential) to the otherterminal (making it negatively charged) (lowerpotential).

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Electric Current

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Electric Current

Is there a limit to the potential differencebetween the two terminals of a battery? Yes.It is called the electromotive force E (nothingto do with a force), e.g. E = 1.5 V for a AAbattery.

How do the conduction electrons move?From low to high potential i.e. from - to +.

(Historical) convention: The direction of thecurrent is taken to be from + to -, opposite tothe direction of the electrons.

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Electric Current

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Electric Current

How many electrons pass through a crosssection of the wire in one second?

Current:

I = ∆ q∆ t

Unit: 1 ampere(A) = 1 C/s

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Electric Current: ExampleA current of 0.2 mA coming from a 3.0 V batteryoperates a calculator for one hour. How muchcharge flows in the circuit?Answer:∆ q = I∆ t = (0.2 × 10−3A)(3600 s) = 0.72 C

Physics 201 – p. 10/34


Electric current

If the current is always in the same direction,you have a direct current or dc current; If thecurrent oscillates, i.e. changes direction, youhave an alternating or ac current.

What is a typical speed of the electrons in acurrent?Answer: A rough calculation indicates that theaverage speed of the electrons called the driftspeed is around 2.4 × 10−4m/s.

Physics 201 – p. 11/34


Electric current

So if I have a wire of length 2.4m, an electronat one end will take 10, 000 s to reach theother end. Why is it that when I flip the switch,the light immediately turns on?Answer: Just because the signal that turns onthe electric field travels at the speed of lightso that all electrons from one end to the othermove at once.

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Ohm’s Law

How much current is flowing inside a circuithooked to a battery? Take a ride on one ofthese electrons. You can actually see that itcollides repeatedly with the atoms of the wire⇒ Resistance to the motion of that electron.

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Ohm’s Law

Any similarity with something that we alreadyknow? Yes. Imagine that you are sliding downa very icy slope. Because of negligiblefriction, most of the potential energy isconverted into kinetic energy. If the slope isvery rough instead, some of that potentialenergy is converted into heat.

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Ohm’s Law

The resistance is translated into arelationship between the applied voltage Vand the current I: Ohm’s Law:

R = VI

R: resistance. Unit: 1 ohm(Ω) = 1 VA

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Ohm’s Law: ExampleThe resistance of a bagel toaster is 14 Ω. Toprepare a bagel, the toaster is operated for oneminute from a 120-V outlet. How much energy isdelivered to the toaster?

Three inputs: R = 14 Ω; t = 60 s; V = 120 V .Concepts?

The energy delivered is equal to the workdone in moving ∆q in ∆t = 60s and by apotential difference of 120 V. ⇒ E = (∆ q) V .

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Ohm’s Law: Example

What’s ∆q?∆q = I∆ t = V

R∆ t

⇒ E = (∆ q) V = V 2

R ∆ t = (120V )2

14Ω (60s) =

6.2 × 104J .

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Resistance and Resistivity

When I am given a piece of conducting wire,how do I know what its resistance might be?Answer: The electrons that travel from oneend to the other encounter more atoms toscatter on as the wire gets longer. Also if theatoms are packed into a smaller area, therewill be more scatterings ⇒ The resistance willget larger.

So?

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Empirical formula for the resistance:

R = ρLA

ρ: Resistivity of the materialL: Length of conducting wireA: Its cross-section

What does that tell us about differentmaterial? Conductors have low resistivity,while insulators have large resistivity. Ingeneral, we want to minimize the resistance.

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Resistance and Resistivity: Some aplications

Impedance (or resistance) plethysmography:Measure the resistance in the calf,R = ρL

A = ρ LVcalf/L = ρ L2

Vcalf. Pressure cuff cuts

off the veinous flow ⇒ Vcalf increases ⇒ Rdecreases. Pressure cuff ⇒ removed ⇒Rapid return to normal resistance if there isno blood clot in the veins. A slow return tonormal indicates some blood clot.

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20-gauge wire’s cross section: 5.2 × 10−7m2;16-gauge wire’s cross section: 13 × 10−7m2.For the same length, the 16-gauge wire hassmaller resistance than the 20-gauge one ⇒less heating (proportional to R) of the wire.

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If I heat up a wire, will its resistance change?Answer: The resistance goes up! Forexample, R = R0(1 + α(T − T0)) where α isthe temperature coefficient of resistivity.

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If I cool the wire to extremely lowtemperatures, what will happen to itsresistance?Answer: There are some material whoseresistance goes to zero as the temperature islowered below some critical temperature Tc.They are called superconductors. Copperoxide complexes such asHg − Ba2Ca2Cu2O8+δ have Tc = 150 K.

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Electrical energy and power

From the example given above, the energydelivered to the toaster is

∆U = ∆qV = IV ∆t

The power is

P = ∆U∆t = IV = I2R = V 2

R

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Electrical energy and power

For the toaster example above, P = 1.03kW

From Eq. (5), one can see that, in order tominimize the power dissipated in terms ofheat, one has to minimize the resistance.

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Resistors in series

What happens to a circuit when I connectresistors in series, i.e. one after the other?Answer: In series means that the samecurrent flows through the resistors.

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Resistors in series

Let me take two resistors, R1 and R2. Can Isimplify the problem ?Answer: Yes. The voltages across theresistors are respectively V1 = IR1 andV2 = IR2. The sum is equal to the emf of thebattery (neglecting internal resistance of thebattery):V = V1 + V2 = IR1 + IR2 = I(R1 + R2) = IReq.Equivalent resistance:

Req = R1 + R2 + ...Physics 201 – p. 28/34


Resistors in series

So does that tell me that for a circuit withresistors in series, I can draw an equivalentcircuit with one resistor whose resistance isthe sum of all the individual resistances?Answer: Yes!

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Resistors in series

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Resistors in parallel

How come all the wall sockets in my househave the same voltage, namely 120 V?Answer: This is an example of a wiring inparallel.

What does it really mean?Answer: In parallel means that the devices(resistors, etc..) are connected in such a waythat the voltage across each one of them isthe same.

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What about the current(s)?Answer: Since I = V/R and V is the same,the one with larger R will have a smallercurrent flowing in it. There will be a current Ii

flowing in each branch i. The sum of all thecurrents in all the branches should be equalto the current produced by the source(battery,etc..)

I = I1 + I2 + I3 + ...

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Can I draw an equivalent circuit?Answer: Yes.I = I1 + I2 + I3 + ... = V

R1

+ VR2

+ .. = VReq

1Req

= 1R1

+ 1R2

+ ..

Physics 201 – p. 33/34



Physics 201 – p. 34/34

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