Question 1

For the circuit shown in the figure, the inductors have no appreciable resistance and the switch has been open for a very long time.  
(a) The instant after closing the switch, what is the current through the 60.0-Ω resistor?
(b) The instant after closing the switch, what is the potential difference across the 15.0-mH inductor?
(c) After the switch has been closed and left closed for a very long time, what is the potential drop across the 60.0-Ω resistor?

Accepted Answer

(a) 1.43 A 
(b) 85.7 V 
(c) 0.00 V

Question 2

What is the energy density in the magnetic field 25 cm from a long straight wire carrying a current of 12 A? (μ₀ = 4π × 10^-7 T ∙ m/A) A) 7.3 × 10^-5 J/m³ B) 3.7 × 10^-5J/m³ C) 3.6 × 10^-4 J/m³ D) 1.2 × 10^-4 J/m³ E) The density cannot be determined without knowing the volume.

Accepted Answer

The magnetic field B at a distance r from a long straight wire is given by B = (μ₀I)/(2πr). The energy density u in a magnetic field is given by u = B²/(2μ₀). Substituting the values, we find the energy density to be approximately 3.7 × 10⁻⁵ J/m³.

Question 3

How much energy is stored in a room 3.0 m by 4.0 m by 2.4 m due to the earth's magnetic field with a strength of 5.0 × 10^-5T? (μ₀= 4π × 10^-7 T ∙ m/A) A) 570 mJ. B) 29 mJ. C) 10 mJ. D) 100 mJ. E) 570 mJ.

Accepted Answer

The energy stored in a magnetic field can be calculated using the equation:

E = 0.5 * μ * V * B^2

Where E is the energy stored, μ is the magnetic permeability of the material in the room (assumed to be air), V is the volume of the room, and B is the strength of the magnetic field.

Substituting the given values, we get:

E = 0.5 * (4π × 10^-7 T∙m/A) * (3.0 m * 4.0 m * 2.4 m) * (5.0 × 10^-5 T)^2
E = 28.8 × 10^-3 J
E = 29 mJ

Therefore, the energy stored in the room due to Earth's magnetic field is 29 mJ. Answer choice B is correct.

Question 4

At a certain instant the current flowing through a 5.0-H inductor is 3.0 A. If the energy in the inductor at this instant is increasing at a rate of 3.0 J/s, how fast is the current changing?
A) 0.20 A/s
B) 0.40 A/s
C) 0.10 A/s
D) 0.80 A/s

Accepted Answer

The answer of At a certain instant the current flowing...

Question 5

At what rate would the current in a 100-mH inductor have to change to induce an emf of 1000 V in the inductor?
A) 100 A/s
B) 1 A/s
C) 1000 A/s
D) 10,000 A/s
E) 10 A/s

Accepted Answer

The induced emf (voltage) in an inductor is given by the formula $V = L \frac{dI}{dt}$, where $V$ is the voltage in volts, $L$ is the inductance in henrys, and $\frac{dI}{dt}$ is the rate of change of current in amperes per second. Rearranging the formula to solve for $\frac{dI}{dt}$, we get $\frac{dI}{dt} = \frac{V}{L}$. Substituting the given values, $V = 1000$ V and $L = 100$ mH = 0.1 H, we find $\frac{dI}{dt} = \frac{1000}{0.1} = 10,000$ A/s.

Question 6

An insulated wire of diameter 1.0 mm and negligible resistance is wrapped tightly around a cylindrical core of radius 5.0 cm and length 30 cm to build a solenoid. What is the energy stored in this solenoid when a current I = 0.20 A flows through it? (μ₀ = 4π × 10^-7 T ∙ m/A) A) 1.2 × 10^-4 J B) 9.6 × 10^-4 J C) 4.8 × 10^-4 J D) 2.4 × 10^-4J E) 5.9 × 10^-5 J

Accepted Answer

The answer of An insulated wire of diameter 1.0 mm...

Question 7

A series LR circuit contains an emf source of 14 V having no internal resistance, a resistor, a 34 H inductor having no appreciable resistance, and a switch. If the emf across the inductor is 80% of its maximum value 4.0 s after the switch is closed, what is the resistance of the resistor?
A) 14 Ω
B) 1.9 Ω
C) 1.5 Ω
D) 5.0 Ω

Accepted Answer

The time constant of the circuit is given by $	au = L/R$, where $L$ is the inductance and $R$ is the resistance. At $t = 0$, the voltage across the inductor is zero, and it increases exponentially towards its maximum value, according to:

$$
V_L = \mathcal{E}(1 - e^{-t/	au})
$$

At $t = 4.0\,	ext{s}$, the voltage across the inductor is $0.8\,\mathcal{E}$. Therefore, we can write:

$$
0.8\,\mathcal{E} = \mathcal{E}(1 - e^{-4.0/	au})
$$

Simplifying this expression, we get:

$$
e^{-4.0/	au} = 0.2
$$

Taking the natural logarithm of both sides, we get:

$$
-\frac{4.0}{	au} = \ln 0.2 \quad \Rightarrow \quad 	au = \frac{4.0}{\ln 5}
$$

Now we can solve for the resistance $R$:

$$
R = \frac{L}{	au} = \frac{34}{4.0/ \ln 5} \approx 1.9\,\Omega
$$

Therefore, the answer is B).

Question 8

A series circuit consists of a 0.55-H inductor with internal resistance of 8.0 Ω connected in series with a 4.0-Ω resistor, an open switch, and an ideal 12-V battery.
(a) When the switch is closed, what is the initial current through the 4.0-Ω resistor?
(b) What is the current through the 4.0-Ω resistor a very long time after the switch is closed?

Accepted Answer

The answer of A series circuit consists of a 0.55-H...

Question 9

For the circuit shown in the figure, the switch has been open for a very long time.  
(a) What is the potential drop across the 15.0-mH inductor just after closing the switch?
(b) What is the potential drop across the 70.0-µF capacitor after the switch has been closed for a very long time?

Accepted Answer

The answer of For the circuit shown in the figure,...

Question 10

What is the self-inductance of a solenoid 30.0 cm long having 100 turns of wire and a cross-sectional area of 1.00 × 10^-4 m2? (μ₀= 4π × 10^-7T ∙ m/A) A) 4.19 nH B) 4.19 pH C) 4.19 µH D) 4.19 mH E) 4.19 H

Accepted Answer

The formula for the self-inductance of a solenoid is L = μn^2Aℓ, where μ is the permeability of free space, n is the number of turns per unit length (n = N/ℓ), A is the cross-sectional area, and ℓ is the length of the solenoid. Substituting the given values, we get:

n = N/ℓ = 100/0.3 = 333.33 turns/m
L = μn^2Aℓ = (4π × 10^-7 T·m/A)(333.33 turns/m)^2(1.00 × 10^-4 m^2)(0.30 m) = 4.19 µH

Therefore, the best choice is C, 4.19 µH.

Question 11

Which of the following statements about inductors are correct? There may be more than one correct choice.
A) When it is connected in a circuit, an inductor always resists having current flow through it.
B) Inductors store energy by building up charge.
C) When an inductor and a resistor are connected in series with a DC battery, the current in the circuit is reduced to zero in one time constant.
D) An inductor always resists any change in the current through it.
E) When an inductor and a resistor are connected in series with a DC battery, the current in the circuit is zero after a very long time.

Accepted Answer

The answer of Which of the following statements about inductors...

Question 12

A series LR circuit consists of a 2.0-H inductor with negligible internal resistance, a 100-ohm resistor, an open switch, and a 9.0-V ideal power source. After the switch is closed, what is the maximum power delivered by the power supply?
A) 0.40 W
B) 81 W
C) 0.090 W
D) 8.1 W
E) 0.81 W

Accepted Answer

The answer of A series LR circuit consists of a...

Question 13

In the figure, the current in a solenoid having no appreciable resistance is flowing from b to a and is decreasing at a rate of 9.6 A/s. The self-induced emf in the solenoid is found to be 8.4 V.  
(a) What is the self-inductance of the solenoid?
(b) Which point, a or b is at higher potential?

Accepted Answer

The answer of In the figure, the current in a...

Question 14

A 45-mH ideal inductor is connected in series with a 60-Ω resistor through an ideal 15-V DC power supply and an open switch. If the switch is closed at time t = 0 s, what is the current 7.0 ms later?
A) 250 mA
B) 650 mA
C) 550 mA
D) 280 mA
E) 850 mA

Accepted Answer

The current in an inductor increases linearly with time when a constant voltage is applied. The rate of increase is determined by the inductance and the voltage. In this case, the current will increase by:```ΔI = (V/L) * t```where:* ΔI is the change in current* V is the voltage* L is the inductance* t is the timePlugging in the values given in the question, we get:```ΔI = (15 V / 45 mH) * 7.0 ms = 225 mA```Therefore, the current 7.0 ms after the switch is closed will be:```I = ΔI + 0 A = 225 mA```which is closest to choice A, 250 mA.

Question 15

An LR circuit contains an ideal 60-V battery, a 42-H inductor having no resistance, a 24-Ω resistor, and a switch S, all in series. Initially, the switch is open and has been open for a very long time. At time t = 0 s, the switch is suddenly closed. How long after closing the switch will the potential difference across the inductor be 24 V?
A) 1.6 s
B) 1.4 s
C) 1.8 s
D) 1.9 s
E) 2.1 s

Accepted Answer

When the switch is first closed, the current through the circuit is zero. As the current starts to flow, the inductor resists the change in current and develops a back EMF that is opposite to the battery voltage. Therefore, the potential difference across the inductor initially is 60 V (battery voltage).

After the switch is closed, the current through the circuit starts to increase and the potential difference across the inductor decreases. The time constant of the circuit is L/R = 42/24 = 7/4 s. After one time constant, the potential difference across the inductor is given by:

V = V0 * e^(-t/tau)

where V0 is the initial potential difference across the inductor (60 V), tau is the time constant (7/4 s), and t is the time elapsed since the switch was closed.

We want to find the time when V is equal to 24 V:

24 = 60 * e^(-t/tau)

e^(-t/tau) = 2/5

-t/tau = ln(2/5)

t = -tau * ln(2/5) = 1.6 s

Therefore, the answer is A.

Question 16

An LR circuit contains an ideal 60-V battery, a 51-H inductor having no resistance, a 21-Ω resistor, and a switch S, all in series. Initially, the switch is open and has been open for a very long time. At time t = 0 s, the switch is suddenly closed. When the voltage across the resistor is equal to the voltage across the inductor, what is the current in the circuit?
A) 1.4 A
B) 0.57 A
C) 1.1 A
D) 0.86 A
E) 1.7 A

Accepted Answer

When the switch is closed, the circuit becomes a series RL circuit. The time constant of this circuit is L/R = 51/21 = 2.43 s.

When the voltage across the resistor is equal to the voltage across the inductor, the current in the circuit is given by:

I = V/R = 60/21 = 2.86 A

The current in the circuit at this point is decreasing exponentially with time, and after one time constant (2.43 s), the current will have decreased to:

I(2.43) = I(0) * e^(-t/tau) = 2.86 * e^(-2.43/2.43) = 1.4 A

Therefore, the correct answer choice is A.

Question 17

What resistance should be added in series with a 3.0-H inductor to complete an LR circuit with a time constant of 4.0 ms?
A) 0.75 k Ω
B) 12 Ω
C) 0.75 Ω
D) 2.5 Ω

Accepted Answer

The time constant τ for an LR circuit is given by τ = L/R. Here, L = 3.0 H and τ = 4.0 ms = 0.004 s. Solving for R gives R = L/τ = 3.0 H / 0.004 s = 750 Ω = 0.75 kΩ.

Question 18

The mutual inductance between two coils is 10.0 mH. The current in the first coil changes uniformly from 2.70 A to 5.00 A in 0.160 s. If the second coil has a resistance of 0.600 Ω, what is the magnitude of the induced current in the second coil?

Accepted Answer

The answer of The mutual inductance between two coils is...

Question 19

A 96-mH solenoid inductor is wound on a form 0.80 m in length and 0.10 m in diameter. A coil is tightly wound around the solenoid at its center. The coil's resistance is 9.9 ohms. The mutual inductance of the coil and solenoid is 31 μH. At a given instant, the current in the solenoid is 540 mA, and is decreasing at the rate of 2.5 A/s. At the given instant, what is the magnitude of the induced current in the coil? (μ₀ = 4π × 10^-7T ∙ m/A) A) 7.8 μA B) 6.3 μA C) 9.4 μA D) 11 μA E) 13 μA

Accepted Answer

The induced emf in the coil is given by:$$emf = -M \frac{di}{dt}$$where:* M is the mutual inductance (31 μH)* di/dt is the rate of change of current in the solenoid (-2.5 A/s)Substituting these values into the equation, we get:$$emf = -(31 \mu H)(-2.5 A/s) = 77.5 \mu V$$The induced current in the coil is then given by:$$i = \frac{emf}{R} = \frac{77.5 \mu V}{9.9 \Omega} = 7.8 \mu A$$Therefore, the magnitude of the induced current in the coil is 7.8 μA.

Question 20

A solenoid of length 0.700 m having a circular cross-section of radius 5.00 cm stores 6.00 µJ of energy when a 0.400-A current runs through it. What is the winding density of the solenoid? (μ₀ = 4π × 10^-7 T ∙ m/A) A) 865 turns/m B) 472 turns/m C) 1080 turns/m D) 104 turns/m E) 327 turns/m

Accepted Answer

The answer of A solenoid of length 0.700 m having...

Quiz 29: Inductance

What is the energy density in the magnetic field 25 cm from a long straight wire carrying a current of 12 A? (μ0 = 4π × 10-7 T ∙ m/A)

How much energy is stored in a room 3.0 m by 4.0 m by 2.4 m due to the earth's magnetic field with a strength of 5.0 × 10-5 T? (μ0 = 4π × 10-7 T ∙ m/A)

At a certain instant the current flowing through a 5.0-H inductor is 3.0 A. If the energy in the inductor at this instant is increasing at a rate of 3.0 J/s, how fast is the current changing?

At what rate would the current in a 100-mH inductor have to change to induce an emf of 1000 V in the inductor?

An insulated wire of diameter 1.0 mm and negligible resistance is wrapped tightly around a cylindrical core of radius 5.0 cm and length 30 cm to build a solenoid. What is the energy stored in this solenoid when a current I = 0.20 A flows through it? (μ0 = 4π × 10-7 T ∙ m/A)

A series LR circuit contains an emf source of 14 V having no internal resistance, a resistor, a 34 H inductor having no appreciable resistance, and a switch. If the emf across the inductor is 80% of its maximum value 4.0 s after the switch is closed, what is the resistance of the resistor?

For the circuit shown in the figure, the switch has been open for a very long time. (a) What is the potential drop across the 15.0-mH inductor just after closing the switch? (b) What is the potential drop across the 70.0-µF capacitor after the switch has been closed for a very long time?

What is the self-inductance of a solenoid 30.0 cm long having 100 turns of wire and a cross-sectional area of 1.00 × 10-4 m2? (μ0 = 4π × 10-7 T ∙ m/A)

Which of the following statements about inductors are correct? There may be more than one correct choice.

A series LR circuit consists of a 2.0-H inductor with negligible internal resistance, a 100-ohm resistor, an open switch, and a 9.0-V ideal power source. After the switch is closed, what is the maximum power delivered by the power supply?

In the figure, the current in a solenoid having no appreciable resistance is flowing from b to a and is decreasing at a rate of 9.6 A/s. The self-induced emf in the solenoid is found to be 8.4 V. (a) What is the self-inductance of the solenoid? (b) Which point, a or b is at higher potential?

A 45-mH ideal inductor is connected in series with a 60-Ω resistor through an ideal 15-V DC power supply and an open switch. If the switch is closed at time t = 0 s, what is the current 7.0 ms later?

What resistance should be added in series with a 3.0-H inductor to complete an LR circuit with a time constant of 4.0 ms?

The mutual inductance between two coils is 10.0 mH. The current in the first coil changes uniformly from 2.70 A to 5.00 A in 0.160 s. If the second coil has a resistance of 0.600 Ω, what is the magnitude of the induced current in the second coil?

A solenoid of length 0.700 m having a circular cross-section of radius 5.00 cm stores 6.00 µJ of energy when a 0.400-A current runs through it. What is the winding density of the solenoid? (μ0 = 4π × 10-7 T ∙ m/A)

What is the energy density in the magnetic field 25 cm from a long straight wire carrying a current of 12 A? (μ₀ = 4π × 10^-7 T ∙ m/A)

How much energy is stored in a room 3.0 m by 4.0 m by 2.4 m due to the earth's magnetic field with a strength of 5.0 × 10^-5T? (μ₀= 4π × 10^-7 T ∙ m/A)

An insulated wire of diameter 1.0 mm and negligible resistance is wrapped tightly around a cylindrical core of radius 5.0 cm and length 30 cm to build a solenoid. What is the energy stored in this solenoid when a current I = 0.20 A flows through it? (μ₀ = 4π × 10^-7 T ∙ m/A)

What is the self-inductance of a solenoid 30.0 cm long having 100 turns of wire and a cross-sectional area of 1.00 × 10^-4 m2? (μ₀= 4π × 10^-7T ∙ m/A)

A solenoid of length 0.700 m having a circular cross-section of radius 5.00 cm stores 6.00 µJ of energy when a 0.400-A current runs through it. What is the winding density of the solenoid? (μ₀ = 4π × 10^-7 T ∙ m/A)