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Physics 212 Lecture 18 Physics 212 Lecture 18, Slide 1 From the prelecture: Self Inductance Wrap a wire into a coil to make an “inductor”… e = -L dI dt Physics 212 Lecture 18, Slide 2 What this really means: emf induced across L tries to keep I constant eL = -L dI dt L current I Inductors prevent discontinuous current changes ! It’s like inertia! Physics 212 Lecture 18, Slide 3 I(t) e = -L dI dt Suppose dI/dt > 0. Induced EMF tries to counteract this change (Lenz’s Law). e + I(t) - V1 V2 + - Voltage across inductor V1 – V2 = VL = + L dI/dt > 0 Checkpoint 1 Two solenoids are made with the same cross sectional area and total number of turns. Inductor B is twice as long as inductor A LB 0 n 2r 2 z (1/2)2 2 LB 1 L A 2 Compare the inductance of the two solenoids A) LA = 4 LB B) LA = 2 LB C) LA = LB D) LA = (1/2) LB E) LA = (1/4) LB Physics 212 Lecture 18, Slide 5 WHAT ARE INDUCTORS AND CAPACITORS GOOD FOR? Inside your i-clicker Physics 212 Lecture 18, Slide 6 How to think about RL circuits Episode 1: When no current is flowing initially: VL I=0 L I=V/R R L t = L/R R I VBATT At t = 0: I=0 VL = VBATT VR = 0 (L is like a giant resistor) VBATT t = L/R At t >> L/R: VL = 0 VR = VBATT I = VBATT/R (L is like a short circuit) Physics 212 Lecture 18, Slide 7 Checkpoint 2a In the circuit, the switch has been open for a long time, and the current is zero everywhere. I At time t=0 the switch is closed. What is the current I through the vertical resistor immediately after the switch is closed? I IL=0 (+ is in the direction of the arrow) A) I = V/R B) I = V/2R C) I = 0 D) I = -V/2R E) I = -V/R Before: IL = 0 After: IL = 0 I = + V/2R Physics 212 Lecture 18, Slide 8 RL Circuit (Long Time) What is the current I through the vertical resistor after the switch has been closed for a long time? (+ is in the direction of the arrow) A) I = V/R B) I = V/2R C) I = 0 D) I = -V/2R E) I = -V/R After a long time in any static circuit: VL = 0 - + + - KVR: VL + IR = 0 Physics 212 Lecture 18, Slide 9 Checkpoint 2b After a long time, the switch is opened, abruptly disconnecting the battery from the circuit. What is the current I through the vertical resistor immediately after the switch is opened? (+ is in the direction of the arrow) A) I = V/R B) I = V/2R C) I = 0 D) I = -V/2R E) I = -V/R circuit when switch opened L IL=V/R R Current through inductor cannot change DISCONTINUOUSLY Physics 212 Lecture 18, Slide 10 Why is there exponential behavior ? I 2 V1 – V2 = L dI dt L 1 VL 3 R t = L/R V3-V4 = IR 4 t = L/R dI L IR 0 dt I (t ) I 0e tR / L I 0e t / t L where t R Physics 212 Lecture 18, Slide 11 I L VL R VBATT t = L/R Lecture: Prelecture: Did we mess up?? No: The resistance is simply twice as big in one case. Physics 212 Lecture 18, Slide 12 Checkpoint 3a After long time at 0, moved to 1 After long time at 0, moved to 2 After switch moved, which case has larger time constant? A) Case 1 B) Case 2 C) The same L t1 2R L t2 3R Physics 212 Lecture 18, Slide 13 Checkpoint 3b After long time at 0, moved to 1 After long time at 0, moved to 2 Immediately after switch moved, in which case is the voltage across the inductor larger? A) Case 1 After switch moved: B) Case 2 V V 2R L1 C) The same Before switch moved: I V R VL 2 R V 3R R Physics 212 Lecture 18, Slide 14 Checkpoint 3c After long time at 0, moved to 1 After long time at 0, moved to 2 After switch moved for finite time, in which case is the current through the inductor larger? A) Case 1 After awhile B) Case 2 I1 Iet / t C) The same 1 Immediately after: I1 I 2 I 2 Iet / t t1 t 2 2 Physics 212 Lecture 18, Slide 15 Calculation The switch in the circuit shown has been open for a long time. At t = 0, the switch is closed. What is dIL/dt, the time rate of change of the current through the inductor immediately after switch is closed R1 V R2 L R3 • Conceptual Analysis – – Once switch is closed, currents will flow through this 2-loop circuit. KVR and KCR can be used to determine currents as a function of time. • Strategic Analysis – – – Determine currents immediately after switch is closed. Determine voltage across inductor immediately after switch is closed. Determine dIL/dt immediately after switch is closed. Physics 212 Lecture 18, Slide 16 Calculation The switch in the circuit shown has been open for a long time. At t = 0, the switch is closed. R1 V R2 L IL = 0 R3 What is IL, the current in the inductor, immediately after the switch is closed? (A) IL =V/R1 up (B) IL =V/R1 down (C) IL = 0 INDUCTORS: Current cannot change discontinuously ! Current through inductor immediately AFTER switch is closed IS THE SAME AS the current through inductor immediately BEFORE switch is closed Immediately before switch is closed: IL = 0 since no battery in loop Physics 212 Lecture 18, Slide 17 Calculation The switch in the circuit shown has been open for a long time. At t = 0, the switch is closed. R1 V R2 L R3 IL(t=0+) = 0 What is the magnitude of I2, the current in R2, immediately after the switch is closed? (A) I2 V R1 (B) I2 V R2 R3 (C) I 2 V R1 R2 R3 (D) I 2 VR2 R3 R2 R3 We know IL = 0 immediately after switch is closed R1 Immediately after switch is closed, V circuit looks like: I I R2 V R1 R2 R3 R3 Physics 212 Lecture 18, Slide 18 Calculation The switch in the circuit shown has been open for a long time. At t = 0, the switch is closed. R1 V R2 L IL(t=0+) = 0 I2 R3 I2(t=0+) = V/(R1+R2+R3) What is the magnitude of VL, the voltage across the inductor, immediately after the switch is closed? (A) VL V R2 R3 R1 (B) VL V (C) VL 0 (D) VL V R2 R3 R2 R3 (E) VL V R1 ( R2 R3 ) R1 R2 R3 Kirchhoff’s Voltage Law, VL-I2 R2 -I2 R3 =0 VL = I2 (R2+R3) VL V R2 R3 R1 R2 R3 Physics 212 Lecture 18, Slide 19 Calculation The switch in the circuit shown has been open for a long time. At t = 0, the switch is closed. What is dIL/dt, the time rate of change of the current through the inductor immediately after switch is closed dI V R2 R3 dI (A) L (B) L 0 dt L R1 dt R1 R2 V L R3 VL(t=0+) = V(R2+R3)/(R1+R2+R3) dI V R2 R3 (C) L dt L R1 R2 R3 dI L V (D) dt L The time rate of change of current through the inductor (dIL /dt) = VL /L dI L V R2 R3 dt L R1 R2 R3 Physics 212 Lecture 18, Slide 20 Follow Up The switch in the circuit shown has been closed for a long time. What is I2, the current through R2 ? (Positive values indicate current flows to the right) (A) I 2 V R2 R3 (B) I 2 V ( R2 R3 ) R1 R2 R3 R1 V R2 L (C) I 2 0 R3 (D) I 2 V R2 R3 After a long time, dI/dt = 0 Therefore, the voltage across L = 0 Therefore the voltage across R2 + R3 = 0 Therefore the current through R2 + R3 must be zero !! Physics 212 Lecture 18, Slide 21 Follow Up 2 The switch in the circuit shown has been closed for a long time at which point, the switch is opened. What is I2, the current through R2 immediately after switch is opened ? (Positive values indicate current flows to the right) (A) I 2 V R1 R2 R3 (B) I 2 V R1 R1 V (C) I 2 0 R2 I2 IL L (D) I 2 R3 V R1 (E) I 2 V R1 R2 R3 Current through inductor immediately AFTER switch is opened IS THE SAME AS the current through inductor immediately BEFORE switch is opened Immediately BEFORE switch is opened: IL = V/R1 Immediately AFTER switch is opened: IL flows in right loop Therefore, IL = -V/R1 Physics 212 Lecture 18, Slide 22