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

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