Physics 2102Physics 2102Lecture 11Lecture 11

DC CircuitsDC Circuits

Physics 2102

Jonathan Dowling

b

a

Incandescent light bulbsIncandescent light bulbs(a) Which light bulb has a smaller resistance: a 60W, or a 100W one?(b) Is the resistance of a light bulb different when it is on and off?(c) Which light bulb has a larger current through its filament: a 60W one, or a

100 W one?(d) Would a light bulb be any brighter if used in Europe, using 240 V outlets?(e) Would a US light bulb used in Europe last more or less time?(f) Why do light bulbs mostly burn out when switched on?

b

a

The battery operates as a “pump” that movespositive charges from lower to higher electricpotential. A battery is an example of an“electromotive force” (EMF) device.

These come in various kinds, and all transform one source of energy into electricalenergy. A battery uses chemical energy, a generator mechanical energy, a solar cellenergy from light, etc.

The difference in potential energy that thedevice establishes is called the EMFand denoted by Ε.

EMF devices and single loop circuitsEMF devices and single loop circuits

Ε = iR

a b c d=a

Va

Ε iR

ba dc− + i

i

Given the emf devices and resistors in a circuit,we want to calculate the circulating currents.Circuit solving consists in “taking a walk” alongthe wires. As one “walks” through the circuit (inany direction) one needs to follow two rules:

When walking through an EMF, add +E if you flow with the current or −Eotherwise. How to remember: current “gains” potential in a battery.

When walking through a resistor, add -iR, if flowing with the current or +iRotherwise. How to remember: resistors are passive, current flows “potential down”.

Example:Walking clockwise from a: + E-iR=0.Walking counter-clockwise from a: − E+iR=0.

Circuit problemsCircuit problems

If one connects resistors of lower and lower value of R to get higher and highercurrents, eventually a real battery fails to establish the potential difference Ε, andsettles for a lower value.One can represent a “real EMF device” as an ideal one attached to a resistor,called “internal resistance” of the EMF device:

The true EMF is a function of current: the morecurrent we want, the smaller Etrue we get.

Ideal batteries vs. real batteriesIdeal batteries vs. real batteries

Etrue = E –i r

E –i r − i R=0 → i=E/(r+R)

Resistors in series and parallelResistors in series and parallelAn electrical cable consists of 100 strands of fine wire, each having 2 Ω resistance. The same potential

difference is applied between the ends of all the strands and results in a total current of 5 A.

(a) What is the current in each strand?Ans: 0.05 A

(b) What is the applied potential difference?Ans: 0.1 V

(c) What is the resistance of the cable?Ans: 0.02 Ω

Assume now that the same 2 Ω strands in the cable are tied in series, one after the other, and the 100times longer cable connected to the same 0.1 Volts potential difference as before.

(d) What is the potential difference through each strand?Ans: 0.001 V

(e) What is the current in each strand?Ans: 0.0005 A

(f) What is the resistance of the cable?Ans: 200 Ω

(g) Which cable gets hotter, the one with strands in parallel or the one with strands in series?Ans: each strand in parallel dissipates 5mW (and the cable dissipates 500 mW);each strand in series dissipates 50 µW (and the cable dissipates 5mW)

…

DC circuits: resistances in seriesDC circuits: resistances in seriesTwo resistors are “in series” if they are connected such that thesame current flows in both.The “equivalent resistance” is a single imaginary resistor that canreplace the resistances in series.

“Walking the loop” results in : E –iR1-iR2-iR3=0 → i=E/(R1+R2+R3)

In the circuit with theequivalent resistance,E –iReq=0 → i=E/Req

Thus,

!=

=n

j

jeq RR1

Multiloop Multiloop circuits: resistors in parallelcircuits: resistors in parallelTwo resistors are “in parallel” if they areconnected such that there is the samepotential drop through both.The “equivalent resistance” is a singleimaginary resistor that can replace theresistances in parallel.

“Walking the loops” results in : E –i1R1=0, E –i2R2=0, E –i3R3=0The total current delivered by the batteryis i = i1+i2+i3 = E/R1+ E/R2+ E/R3.In the circuit with the equivalent resistor,i=E/Req. Thus,

!=

=n

j jeq RR 1

11

Resistors and CapacitorsResistors and Capacitors

Resistors Capacitors

Key formula: V=iR Q=CV

In series: same current same charge Req=∑Rj 1/Ceq= ∑1/Cj

In parallel: same voltage same voltage 1/Req= ∑1/Rj Ceq=∑Cj

AV

R

Vi 5.1

8

12=

!==

ExampleExample

Bottom loop: (all else is irrelevant)

8Ω12V

Which resistor gets hotter?

ExampleExample

a) Which circuit has thelargest equivalentresistance?

b) Assuming that allresistors are thesame, which onedissipates morepower?

c) Which resistor hasthe smallest potentialdifference across it?

ExampleExampleFind the equivalent resistance between points(a) F and H and(b) F and G.(Hint: For each pair of points, imagine that a batteryis connected across the pair.)

Monster MazesMonster MazesIf all resistors have aresistance of 4Ω, and allbatteries are ideal andhave an emf of 4V,what is the currentthrough R?

If all capacitors have acapacitance of 6µF, andall batteries are idealand have an emf of10V, what is the chargeon capacitor C?