Circuits. David J. Starling Penn State Hazleton PHYS 212

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1 Invention is the most important product of man s creative brain. The ultimate purpose is the complete mastery of mind over the material world, the harnessing of human nature to human needs. - Nikola Tesla RC David J. Starling Penn State Hazleton PHYS 212

2 A battery does work on charge in a circuit. How much work does it do, per charge? This is called the electromotive force, or emf. Definition: E = dw dq (1) RC Examples: batteries, generators, solar cells, fuel cells (H), electric eels, human nervous system

3 Let s look at the energy in this circuit: RC The battery does work: dw = Edq The resistor dissipate energy: Pdt = i 2 Rdt Recall the definition of current: dq = idt

4 Let s look at the energy in this circuit: RC Putting this together: dw = Pdt (conservation of energy) E dq = i 2 R dt Ei = i 2 R E = ir The emf E of an emf device is the potential V that it produces.

5 Lecture Question 10.1: Consider a circuit that contains an ideal battery and a resistor to form a complete circuit. Which one of the following statements concerning the work done by the battery is true? (a) No work is done by the battery in such a circuit. (b) The work done is equal to the thermal energy dissipated by the resistor. (c) The work done is equal to the work needed to move a single charge from one side of the battery to the other. (d) The work done is equal to the emf of the battery. (e) The work done is equal to the product ir. RC

6 A few rules to remember: Resistance Rule: the potential drops by ir across a resistor in the direction of the current; the potential increases by ir in the opposite direction. Emf Rule: the potential increases by E from the negative to the positive terminal, regardless of the direction of current. Loop Rule: the sum of the changes in potential encountered in a complete traversal of any loop of a circuit must be zero. Junction Rule: the sum of the currents entering a junction equals the sum of the currents leaving that junction. RC

7 Resistance Rule The potential drops by ir across a resistor in the direction of the current; the potential increases by ir in the opposite direction. RC V f V i = ir V = ir Or, V i V f = +ir (against the current)

8 Emf Rule The potential increases by E from the negative to the positive terminal, regardless of the direction of current. RC The roles of V i and V f have switched. V i V f = E

9 Loop Rule The sum of the changes in potential encountered in a complete traversal of any loop of a circuit must be zero. RC Start at V i and end at V i : V i ir + E = V i Opposite direction: V i E + ir = V i. Both equations: E ir = 0

10 Junction Rule The sum of the currents entering a junction equals the sum of the currents leaving that junction. RC For junction b: i 2 = i 1 + i 3 For junction d: i 1 + i 3 = i 2

11 Let s look at a real battery: RC Most batteries have some internal resistance r. This resistance makes the battery voltage seem smaller than the emf E. The larger the current, the worse the effect.

12 Let s look at a real battery: RC Let s sum up the changes: E ir ir = 0 i = E r + R

13 Another example: RC Let s sum up the changes: E ir 1 ir 2 ir 3 = 0 i = E R 1 + R 2 + R 3 This is the same if we replace all three series resistors with R eq = R 1 + R 2 + R 3.

14 For resistors in series: RC We replace them with an equivalent resistance: R eq = R 1 + R 2 + R 3 = 3 R i (2) i=1

15 Lecture Question 10.2 Which resistors, if any, are connected in series? RC (a) R 1 and R 2 (b) R 1 and R 3 (c) R 2 and R 3 (d) R 1, R 2 and R 3 (e) None

16 Three parallel resistors: RC Resistors in parallel all have the same voltage V But they may have different currents In this example, i = i 1 + i 2 + i 3

17 Three parallel resistors: RC Each current is given by i 1 = V/R 1 This gives: i = i 1 +i 2 +i 3 = V R 1 + V R 2 + V R 3 = V ( ) R 1 R 2 R 3

18 For three parallel resistors, we can use an equivalent resistance: RC such that V = ir eq, with 1 R eq = 1 R R R 3

19 In summary: RC

20 RC Consider a circuit with a resistor and a capacitor RC With the battery connected, we find: E ir q/c = 0 (loop rule) i = dq/dt This gives: R dq dt + q C = E

21 RC R dq dt + q C = E This is a differential equation RC The solution, with q(0) = 0: q(t) = CE(1 e t/rc ) Don t believe me? Let s check it! Note: V = q/c = E(1 e t/rc )

22 RC The combination τ = RC is the capacitive time constant. If t = τ, then q(τ) = CE(1 e RC/RC ) = 0.63CE RC τ is the time it takes for the capacitor to charge about 63%

23 RC Once the capacitor is charged, let s remove the battery: RC With the battery bypassed, we find: ir q/c = 0 (loop rule) i = dq/dt This gives: R dq dt + q C = 0

24 RC R dq dt + q C = 0 The solution, with q(0) = CE: RC q(t) = CEe t/rc The battery discharges with the same time constant!

25 RC Lecture Question 10.4 For an RC circuit, how long does it take (in theory) for the capacitor to charge from 0% to 100%? RC (a) 0 s (b) 63 s (c) RC (d) 1 RC (e) Infinitely long (it never charges completely)

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