B5, General Physics Experiment II Fall Semester, 2020kpope.korea.ac.kr/GenPhyLab/Manual/B5.pdf ·...
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B5, General Physics Experiment II τ RL Fall Semester, 2020 Name: Team No. : Department: Date : Student ID: Lecturer’s Signature : Introduction Goals • Measure the inductive time constant (τ RL ). • Understand the inductive reactance (X L ). • Understand the phase shift between current and potential difference. Theoretical Backgrounds 1. Potential Difference (a) Faraday’s law of induction states that the emf of a circuit is generated by E = - dΦ B dt , if the magnetic field flux, Φ B = Z B · dA, varies with time. (b) The inductance L is constant if the geometric structure of an inductor is stationary. In that case, L =Φ B /I . (c) The potential difference v L between the two terminals of an inductor of inductance L is V L = L di(t) dt . 2. Inductance Consider a solenoid of length ‘, cross sectional area A, and the number of turns N . (a) Ampere’s law can be used to compute the flux passing through the solenoid as Φ B = μ 0 NIA ‘ , where I is the current and μ 0 is the permeability of vacuum. (b) The inductance is given by L = μ 0 N 2 A ‘ . (c) It is straightforward to derive the formula for the equivalent inductance for a serial connection of N inductors L i ’s: L = N X i=1 L i . (d) The equivalent inductance for a parallel connection of N inductors L i ’s is: 1 L = N X i=1 1 L i . 3. Kirchhoff’s Rule The algebraic sum of the changes in potential during a travel over a closed loop vanishes. The formula is N X i=1 ΔV i =0, where i indicates each circuit element on that closed path. 4. RL circuit with a square wave input (a) The potential equation is v L + v R = E , where E is the potential difference between the two terminals of the battery. (b) This is the first-order linear differential equation: L di dt + Ri = E . (c) If I (t = 0) = 0, then v L (t) = L di dt = E e -t/τ RL , v R (t) = Ri = E (1 - e -t/τ RL ), where τ RL is the time constant, τ RL = L R . © 2020 KPOPE All rights reserved. Korea University Page 1 of 5
Transcript of B5, General Physics Experiment II Fall Semester, 2020kpope.korea.ac.kr/GenPhyLab/Manual/B5.pdf ·...
B5, General Physics Experiment II τRL Fall Semester, 2020
Name:
Team No. :
Department:
Date :
Student ID:
Lecturer’s Signature :
Introduction
Goals
• Measure the inductive time constant (τRL).
• Understand the inductive reactance (XL).
• Understand the phase shift between current andpotential difference.
Theoretical Backgrounds
1. Potential Difference
(a) Faraday’s law of induction states that the emfof a circuit is generated by
E = −dΦB
dt,
if the magnetic field flux,
ΦB =
∫B · dA,
varies with time.
(b) The inductance L is constant if the geometricstructure of an inductor is stationary. In thatcase, L = ΦB/I.
(c) The potential difference vL between the twoterminals of an inductor of inductance L is
VL = Ldi(t)
dt.
2. Inductance
Consider a solenoid of length `, cross sectional areaA, and the number of turns N .
(a) Ampere’s law can be used to compute the fluxpassing through the solenoid as
ΦB = µ0NIA
`,
where I is the current and µ0 is thepermeability of vacuum.
(b) The inductance is given by
L = µ0N2A
`.
(c) It is straightforward to derive the formula forthe equivalent inductance for a serialconnection of N inductors Li’s:
L =
N∑i=1
Li.
(d) The equivalent inductance for a parallelconnection of N inductors Li’s is:
1
L=
N∑i=1
1
Li.
3. Kirchhoff’s Rule
The algebraic sum of the changes in potentialduring a travel over a closed loop vanishes. Theformula is
N∑i=1
∆Vi = 0,
where i indicates each circuit element on thatclosed path.
4. RL circuit with a square wave input
(a) The potential equation is
vL + vR = E ,
where E is the potential difference betweenthe two terminals of the battery.
(b) This is the first-order linear differentialequation:
Ldi
dt+Ri = E .
(c) If I(t = 0) = 0, then
vL(t) = Ldi
dt= E e−t/τRL ,
vR(t) = Ri = E (1 − e−t/τRL ),
where τRL is the time constant,
τRL =L
R.
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B5, General Physics Experiment II τRL Fall Semester, 2020
(d) If I(0) = I0 [dIdt (t = 0) = 0] and we set E = 0at t = 0, then
vL(t) = LdI
dt= −RI0e−t/τRL ,
vR(t) = RI = RI0e−t/τ
RL .
5. RL circuit with a sine wave input
(a) This is the first-order linear differentialequation with a sine wave input:
vR + vL = Ri+ Ldi
dt= Em sinωdt,
where ωd (Em) is the frequency (amplitude) ofthe sine wave input.
(b) The current i(t) of the circuit is
i(t) =EmZ
sin(ωdt− φ),
where Z is
Z =√R2 +X2
L,
XL is the inductive reactance
XL = ωdL,
and φ is the phase constant
φ = arctanXL
R.
(c) The potential difference vR(t) of R is
vR(t) = i(t)R
=EmR
Zsin(ωdt− φ).
The phase of i(t) and that of vR(t) are inphase.
(d) The potential difference vL(t) of L is
vL(t) = Ldi(t)
dt
=LωdEmZ
cos(ωdt− φ)
=LωdEmZ
sin(ωdt− φ+π
2).
The phase of vL(t) leads that of i(t) by π2 , or
the phase of i(t) lags that of vL(t) by π2 .
(e) The phase of vL(t) leads that of vR(t) by π2 ,
or the phase of vR(t) lags that of vL(t) by π2 .
Instrumentation
1. Board RL
2. Connection
Experimental Procedure
1. E1: a square wave input
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B5, General Physics Experiment II τRL Fall Semester, 2020
(a) Carry out experiment with the
five circuits on the borad RL.
(b) Connect the signal generator to Vin and theground of the board.
(c) Connect the voltage sensor across the groundand Vin. Also connect the voltage sensoracross the resistor and the inductor.
(d) Set the scope to watch each potentialdifference.
(e) Set the waveform of the signal generator to
Positive Square Wave .
(f) Set the frequency and amplitude of the signalgenerator to 500 Hz and 1 V. Also checkAuto of the signal generator.
(g) Set the scope to watch each potentialdifference.
(h) Set the trigger of the scope.
(i) Select Fast Monitor Mode and click Monitorto observe.
(j) Open the graph and plot the potentialdifference across the resistor and the inductor.
(k) Click Highlight range of points in active dataand fit the data to
A+Be(−t/τ).
2. E2: a sine wave input
(a) Carry out experiments with a circuit on theborad RL.
(b) Connect the signal generator to Vin and theground of the board.
(c) Connect the voltage sensor across the groundand Vin. Also connect the voltage sensoracross the resistor and the inductor.
(d) Set the scope to watch each potentialdifference.
(e) Set the waveform of the signal generator to
Sine .
(f) Set the frequency and amplitude of the signalgenerator to 500 Hz and 1 V. Also checkAuto of the signal generator.
(g) Set the scope to watch each potentialdifference.
(h) Set the trigger of the scope.
(i) Select Fast Monitor Mode and click Monitorto observe.
(j) Measure the phase difference between vR andvL.
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B5, General Physics Experiment II τRL Fall Semester, 2020
Name:
Team No. :
Department:
Date :
Student ID:
Lecturer’s Signature :
Discussion (7 points)
Problem 1Problem 2
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B5, General Physics Experiment II τRL Fall Semester, 2020
Problem 3
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