Electromagnetism Physics 5b Lecture #9 DC Circuits Purcell 4.7 4. What We Did Last Time Define current and current density J J da S Charge conservation divj ρ or 0 in steady current t Carrier densities n k, velocities u k J n k q k u k J σe Ohm s Law (micro) (macro) For constant cross section A, Physics of electrical conduction R R L σ A Assuming that carrier (electron) motion is dominated by thermal fluctuation k σ n e e2 τ m e
Today s Goals One more resistance problem How to go between micro (resistivity) and macro (resistance) Power dissipation in a resistor Discuss direct current (DC) circuits Made of resistors, capacitors and electromotive forces Resistor arithmetic Many of you already know Kirchhoff s rules Simple and straightforward rules that help you to solve DC circuit problems systematically Gustav Kirchhoff (824 887) Cylindrical Resistor A material of resistivity ρ is filled between two highlyconductive tubes of radii a and b (a < b), length Apply potential difference between tubes E and J are both pointing out At radius r, J(r) σe(r) ρ E(r)ˆr ntegrate J over a cylinder at r J(r) da 2πrLJ(r) E(r) ρj(r) ρ 2πrL ˆr cylinder ntegrate E from a to b b E(r) dr ρ b dr a 2πL ρ a r 2πL ln b a R ρ 2πL ln b a a r J b 2
Power Dissipation Current flows through a piece of resistive wire Every t seconds, charge Q t moves across potential difference They lose energy U Q t What happens to the energy? Current flowing downstream across dissipates power and turn it into heat P 2 R Unit: erg/sec in CGS, joule/sec watt (W) in S q q q Next: We need a source of energy to keep it going R Electromotive Force Batteries supply electricity Connect it to a resistor and we ll find and Connect it to a capacitor and we ll find and Q R Q C Q A battery is a potential-difference generator We call a battery s ability to produce a potential difference an electromotive force, or emf Unit: statvolt in CGS, volt in S Watch out! t s not a force 3
What is an EMF nside a battery, chemical reactions move the charges from one contact to another t s like a pump pushing water up R h Battery converts chemical energy to potential energy of charges Pump converts kinetic energy to potential energy of water nternal Resistance Connect a battery with an emf E to a resistor R The current should be E R E! True only for an ideal battery R Real batteries have internal resistance E! nternal resistance deal battery measured outside the battery is E < E R E! Loss due to internal resistance 4
Measuring E and nternal loss,, is proportional to the current We can measure with no current and get E! This is open-circuit voltage We can measure with current and find out E E! E! R E! R ER R Battery testers do this to determine if is too large Dead batteries have the same emf E as the fresh ones, but larger internal resistance Series/Parallel Resistors Series Parallel Current through and are the same Potential differences across and are and 2 Total potential must be 2 R equiv. Potential across and are the same Currents through and are / and 2 / Combined, they draw 2 R equiv. R equiv. R equiv. 5
Resistor Arithmetic For arbitrary number of resistors R n R n R equiv. i R Generally, any 2-terminal network equiv. of resistors is equivalent to a single resistor t may be easy or not so easy i Not so easy ones need a strategy R 4 Kirchhoff s Rules #) At any point in a circuit, the sum of the incoming current equals to the sum of the outgoing current in Junction Rule out #2) Around any loop in a circuit, the sum of the potential differences is zero 0 Loop Rule loop These are restatements of what we know perfectly well Electric charge is conserved d Current is divergent free J 0 ρ dv dt 0 Electric potential is conservative Potential is curl free E 0 E ds 0 C 6
Using Kirchhoff Let s do a simple example: Find the current through Step : Define currents One for each branch of the circuit Direction doesn t matter Pick one Step 2: Apply the junction rule Junction A 2 A B E Junction B 2 Step 3: Apply the loop rule Top loop E 0 Bottom loop 2 0 2 What s this minus sign doing here? Loop Rule Subtlety To apply the loop rule, you must decide which direction to go around each loop Either will work, just pick one As you go around the loop, An emf counts as E if you pass through it from to A resistor counts as f the current is flowing the direction you are going f you are going this way A 2 B E E E R R 2 0 7
Back to the Problem We ve found so far Junctions: 2 Top loop: E 0 Bottom loop: 2 0 We introduced 3 unknown currents and found 3 equations Solve them: ( )E 2 E ( ) A 2 B E E Kirchhoff Strategy Step : Define currents One for each branch of the circuit Pick the direction of the current Step 2: Apply the junction rule Step 3: Apply the loop rule Pick the direction of the loop Pay attention to the polarity loop You ll have just the right number of equations to solve for all the currents you defined in Step The rest is straightforward (it may be tedious) R E E R 8
Thévenin/Norton Equivalences Any 2-terminal circuit of emfs and resistors can be replaced with an emf and a resistor in series, or a current source and a resistor in parallel Thévenin Norton R Th E Th! No! R No They cannot be electrically distinguished from outside They have the same - relation on the terminals The equivalence theorems assume linearity (including reversibility) of the emfs and resistors Symbol for a current source Finding Equivalences Two parameters Two points on the - relation Open-circuit voltage O : potential difference between the terminals with nothing connected Short-circuit current S : current between the terminals when they are shorted O E Th, S E Th R Th for Thévenin O No R No, S No for Norton R Th E Th! Th O, No S, R Th R No O S Let s do an example No! R No 9
Example Thévenin/Norton Put the circuit we looked at before in a black box Open circuit : we have already solved for, 2, and R 0 3 ( E ) Short circuit : connecting A and B means adding 0Ω in parallel with Effectively this makes 0 will become S S R3 0 E A 2 E B We find: R eq R o S Equivalent Resistance Thévenin/Norton equivalent resistance turns out to be R eq R 3 One can find R eq by replacing all emfs with 0Ω and calculating the total resistance Consider the differential - relation d For a resistor d(r) R d d For an emf d d de d 0 i.e., an emf s differential resistance is 0 A 0Ω 0Ω B 0
Summary Power dissipation in resitors Electromotive forces (emfs) Batteries are made of an emf and an internal resistance Resistor arithmetic Kirchhoff s rules in out R series i Junction Rule P 2 R R parallel 0 loop i Loop Rule Loop rule requires attention to the polarity Tévenin and Norton equivalences Open-circuit voltage and short-circuit current R eq can be evaluated by emf 0Ω Th O No S R Th R No O S