PHYSICS 1E03 Chapter Notes - Chapter 32: Rl Circuit, Ampere, Inductance

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18 Apr 2017

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Self-induction and inductance: distinguish between emfs and currents that are caused by physical sources such as batteries and those that are caused by changing magnetic fields. Emf / current without and adjective describes parameters associated with a physical source. Induced emfs/ currents caused by a changing magnetic field: consider this circuit, consisting of a switch, resistor, and a source of emf: After the switch is closed, the current produces a magnetic flux through the area enclosed by the loop. As the current increases toward its equilibrium value, this magnetic flux changes in time and induces an emf in the loop. The direction of the induced emf is opposite the direction of the emf of the battery, which results in a gradual rather than instantaneous increase in the current to its final equilibrium value. Effect called self-induction because the changing flux through the circuit and the resultant induced emf arise from the circuit itself.

Related Questions

You are given a circular conducting wire loop with one
end cut out and connected to a resistor, R, as shown in
the figure to the right. A metal rod of length l is set at an
initial angle ?0 from the resistor, and then rotates
clockwise about the center of the wire loop with angular
frequency through a uniform magnetic field B
directed into the page. The metal rod stays in contact
with the wire loop at all points of its motion. Unless
otherwise requested, all answers should be in simplified
algebraic form using the above given variables.

(a) Given that the initial angle ?0 is in radians, what is theinitial area of the closed circuit âwedgeâdefined by ?0? (2pts.)
(b) What is the magnetic flux through this initial âwedgeâ areadefined by ?0? (2 pts.)
Starting at time t=0 at position ?0, the rod is rotated clockwisewith angular velocity as shown in
the figure. Given this configuration, answer questions (c)-(g)below:
(c) What is the direction of the induced magnetic field inside theâwedgeâ? (2 pts)
(d) What is the direction of the induced current around the closedcircuit? (2 pts)
(e) What is the magnitude of the induced emf, E in the circuit? (4pts)
(f) What is the magnitude of the induced current in the circuit? (4pts)
(g) What is the power used by the resistor? (4 pts)
(h) The metal rod is now rotated in the opposite direction, i.e. inthe counter-clockwise direction. Given this information, completethe chart below as to what quantities you just calculated change,and what ones do not change. You do not have to show work, justgive the result. No partial credit will be given here. (5pts)

Direction of induced magnetic field inside the âwedgeâ.
Direction of the induced current around the closed circuit.
Magnitude of the induced emf, E in the circuit.
Magnitude of the induced current in the circuit.
Power used by the resistor.

The setup is now modified slightly so that the circular conductingwire loop no longer has a small
section removed as shown in both figures below. This modificationwould allow the metal rod to
continuously rotate on the circular wire loop as clearlyillustrated in the 3D side view. The metal rod
is rotated clockwise, as in the initial part of the problem.

(i) Given this information, complete the chart below. You do nothave to show work, just give the
result. No partial credit will be given here. (5 pts)

Direction of induced magnetic
field inside the âwedgeâ.
Direction of the induced current
around the closed circuit.
Magnitude of the induced emf, E in
the circuit.
Magnitude of the induced current
in the circuit.
Power used by the resistor.

In a circuit containing only resistors, the basic (though notnecessarily explicit) assumption is that the current reaches itssteady-state value instantly. This is not the case for a circuitcontaining inductors. Due to a fundamental property of an inductorto mitigate any "externally imposed" change in current, the currentin such a circuit changes gradually when a switch is closed oropened.
Consider a series circuit containing a resistor of resistance andan inductor of inductance connected to a source of emf andnegligible internal resistance. The wires (including the ones thatmake up the inductor) are also assumed to have negligibleresistance.
Let us start by analyzing the process that takes place after switchis closed (switch remains open). In our further analysis, lowercaseletters will denote the instantaneous values of various quantities,whereas capital letters will denote the maximum values of therespective quantities.
Note that at any time during the process, Kirchhoff's loop ruleholds and is, indeed, helpful:
.
Part A
Immediately after the switch is closed, what is the current in thecircuit?
Hint A.1
How to approach the problem
Hint not displayed
ANSWER:

zero
Answer not displayed
Part B
Immediately after the switch is closed, what is the voltage acrossthe resistor?
Hint B.1
Ohm's law
Hint not displayed
ANSWER:

zero
Answer not displayed
Part C
Immediately after the switch is closed, what is the voltage acrossthe inductor?
Hint C.1
A formula for voltage across an inductor
Hint not displayed
ANSWER:

zero
Answer not displayed
Part D
Shortly after the switch is closed, what is the direction of thecurrent in the circuit?
ANSWER:
clockwise
counterclockwise
There is no current because the inductor does not allow the currentto increase from its initial zero value.
Answer not displayed
Part E
Shortly after the switch is closed, what is the direction of theinduced EMF in the inductor?
ANSWER:
clockwise
counterclockwise
There is no induced EMF because the initial value of the current iszero.
Answer not displayed
Part F
Eventually, the process approaches a steady state. What is thecurrent in the circuit in the steady state?
ANSWER:

Answer not displayed
Part G
What is the voltage across the inductor in the steady state?
Hint G.1
Induced EMF and current
Hint not displayed
ANSWER:
zero

Answer not displayed
Part H
What is the voltage across the resistor in the steady state?
ANSWER:
zero

Answer not displayed
Part I
Now that we have a feel for the state of the circuit in its steadystate, let us obtain the expression for the current in the circuitas a function of time. Note that we can use the loop rule (goingaround counterclockwise):
.
Note as well that and . Using these equations, we can get, aftersome rearranging of the variables and making the subsitution,
.
Integrating both sides of this equation yields
.
Use this last expression to obtain an expression for . Rememberthat and that .
Express your answer in terms of , , and . You may or may not needall these variables. Use the notation exp(x) for .
ANSWER:
=
Answer not displayed
Part J
Just as in the case of R-C circuits, the steady state here is neveractually reached: The exponential functions approach their limitsasymptotically as . However, it usually does not take very long forthe value of to get very close to its presumed limiting value. Thenext several questions illustrate this point.
Note that the quantity has dimensions of time and is called thetime constant (you may recall similar terminology applied to R-Ccircuits). The time constant is often denoted by . Using , one canwrite the expression

as
.
Find the ratio of the current at time to the maximum current.
Express your answer numerically, using three significantfigures.
ANSWER:
=
Answer not displayed
Part K
Find the time it takes the current to reach 99.999% of its maximumvalue.
Express your answer numerically, in units of . Use threesignificant figures.
ANSWER:
=
Answer not displayed

Part L
Find the time it takes the current to reach 99.999% of its maximumvalue. Assume that ohms and millihenrys.
Express your answer in seconds, using three significantfigures.
ANSWER:
=
Answer not displayed
seconds
Part M
Find the time it takes the current to reach 99.999% of its maximumvalue. Assume that ohms and henrys.
Express your answer in seconds, using three significantfigures.
ANSWER:
=
Answer not displayed
seconds
Part N
What fraction of the maximum value will be reached by the currentone minute after the switch is closed? Again, assume that ohms andhenrys.
Use three significant figures in your answer.
ANSWER:
=
Answer not displayed
Now consider a different situation. After switch has been closedfor a long time, it is opened; simultaneously, switch is closed, asshown in the figure. This effectively removes the battery from thecircuit.
The questions below refer to the time immediately after switch isopened and switch is closed.
Part O
What is the direction of the current in the circuit?
ANSWER:
clockwise
counterclockwise
The current is zero because there is no EMF in the circuit.
Answer not displayed
Part P
What is happening to the magnitude of the current?
ANSWER:
The current is increasing.
The current is decreasing.
The current remains constant.
Answer not displayed
Part Q
What is the direction of the EMF in the inductor?
ANSWER:
clockwise
counterclockwise
The EMF is zero because the current is zero.
The EMF is zero because the current is constant.
Answer not displayed
Part R
Which end of the inductor has higher voltage (i.e., to which end ofthe inductor should the positive terminal of a voltmeter beconnected in order to yield a positive reading)?
Hint R.1
How to approach the problem
Hint not displayed
ANSWER:
left
right
The potentials of both ends are the same.
The answer depends on the magnitude of the time constant.
Answer not displayed
Part S
For this circuit, Kirchhoff's loop rule gives
.
Note that , since the current is decreasing. Use this equation toobtain an expression for .
Hint S.1
Initial and final conditions
Hint not displayed
Hint S.2
Let us cheat a bit
Hint not displayed
Express your answer in terms of , , and . Use exp(x) for .
ANSWER:
=
Answer not displayed

PHYSICS 1E03 Chapter Notes - Chapter 32: Rl Circuit, Ampere, Inductance

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