Question 1

A 2.0-μF capacitor that is initially uncharged is charged through a 50-kΩ resistor. How long does it take for the capacitor to reach 90% of its full charge?

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Question 2

In the circuit shown in the figure, all the capacitors are initially uncharged when the switch S is suddenly closed, and the battery is ideal. Find (a) the maximum reading of the ammeter and (b) the maximum charge on the 5.00-µF capacitor.

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Question 3

A multiloop circuit is shown in the figure, but some quantities are not labeled. Find the current I₂ if the batteries are ideal. (It is not necessary to solve the entire circuit.)

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Question 4

For the circuit shown in the figure, V = 20 V, C = 10 µF, R = 0.80 MΩ, and the battery is ideal. Initially the switch S is open and the capacitor is uncharged. The switch is then closed at time t = 0.00 s. What is the potential difference across the resistor 20 s after closing the switch?

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Question 5

The capacitor shown in the circuit in the figure is initially uncharged when the switch S is suddenly closed, and the battery is ideal. After one time constant has gone by, find (a) the current through the resistor and (b) the charge on the capacitor. Assume that the numbers shown are all accurate to two significant figures.

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Question 6

For the circuit shown in the figure, C = 13 µF and R = 7.6 MΩ. Initially the switch S is open with the capacitor charged to a voltage of 80 V. The switch is then closed at time t = 0.00 s. What is the charge on the capacitor 40 s after closing the switch?

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Question 7

A 1.0-μF capacitor is charged until it acquires a potential difference of $900.0 \mathrm {~V}$ across its plates, and then the emf source is removed. If the capacitor is then discharged through a $500.0 - \mathrm { k } \Omega$ resistance, what is the voltage drop across the capacitor $9.0 \mathrm {~ms}$ after beginning the discharge?

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Question 8

A resistor with a resistance of 360 Ω is in a series circuit with a capacitor of capacitance 7.3 × 10^-6 F. What capacitance must be placed in parallel with the original capacitance to change the capacitive time constant of the combination to three times its original value?

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Question 9

A circuit contains a 2.0-MΩ resistor in series with an uncharged capacitor. When this combination is connected across an ideal battery, the capacitor reaches 25% of its maximum charge in 1.5 s. What is its capacitance?

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Question 10

A 2.0-μF capacitor is charged to 12 V and then discharged through a 4.0-MΩ resistor. How long will it take for the voltage across the capacitor to drop to 3.0 V?

Accepted Answer

The time constant, τ, of the circuit can be found using the formula τ = RC, where R is the resistance and C is the capacitance. 
So, τ = (4.0 x 10^6 Ω) x (2.0 x 10^-6 F) = 8.0 s. 
The voltage across the capacitor can be calculated using the formula V = V0 e^(-t/τ), where V0 is the initial voltage (12 V), t is the time, and e is the mathematical constant approximately equal to 2.718. 
Setting V = 3 V and solving for t, we get t = -ln(0.25) x τ = 11 s (rounded to the nearest whole number). Therefore, the best choice is (B) 11 s.

Question 11

A fully charged 37-µF capacitor is discharged through a 1.0-kΩ resistor. If the voltage across the capacitor is reduced to 7.6 volts after just 20 ms, what was the original potential across the capacitor?

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Question 12

A series circuit consists of a 2.5-μF capacitor, a 7.6-MΩ resistor, and an ideal 6.0-V dc power source.
(a) What is the time constant for charging the capacitor?
(b) What is the potential difference across the capacitor 25 s after charging begins?

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Question 13

For the circuit shown in the figure, V = 60 V, C = 20 µF, R = 0.10 MΩ, and the battery is ideal. Initially the switch S is open and the capacitor is uncharged. The switch is then closed at time t = 0.00 s. What is the charge on the capacitor 8.0 s after closing the switch?

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Question 14

For the circuit shown in the figure, V = 60 V, C = 40 µF, R = 0.90 MΩ, and the battery is ideal. Initially the switch S is open and the capacitor is uncharged. The switch is then closed at time t = 0.00 s. At a given instant after closing the switch, the potential difference across the capacitor is twice the potential difference across the resistor. At that instant, what is the charge on the capacitor?

Accepted Answer

We can use the formula for the potential difference across a charging capacitor: 
$V_C=V_0(1-e^{-t/RC})$
where $V_0$ is the initial voltage across the capacitor (in this case, 0 V), $t$ is the time since the switch was closed, $R$ is the resistance, and $C$ is the capacitance.

We also know that at the instant in question, $V_C=2V_R$. We can use Ohm's Law to find $V_R$: 
$V_R=IR$
where $I$ is the current. We can use Kirchhoff's Loop Rule to find the current: 
$-V+V_R+V_C=0$
Plugging in the given values, we get: 
$-60+IR+2IR=0$
$3IR=60$
$I=20	ext{ }\mu	ext{A}$

Now we can go back to the equation for $V_C$: 
$2V_R=V_C$
$2IR=V_0(1-e^{-t/RC})$
$2(20	ext{ }\mu	ext{A})(0.9	ext{ M}\Omega)=0(1-e^{-t/(40	ext{ }\mu	ext{F})(0.9	ext{ M}\Omega)})$
Simplifying, we get: 
$t=\ln 2\cdot RC=12.4	ext{ s}$

Now we can use the formula for the charge on a capacitor: 
$Q=CV$
$Q=40	ext{ }\mu	ext{F}(60	ext{ V})$
$Q=2400	ext{ }\mu	ext{C}$

However, we need to find the charge on the capacitor at the instant when $V_C=2V_R$, which occurs some time after the switch is closed. At that instant, we can use the formula for $V_C$ again: 
$V_C=V_0(1-e^{-t/RC})$
$2V_R=V_C$
$2IR=V_0(1-e^{-t/RC})$
$2(20	ext{ }\mu	ext{A})=0.9	ext{ M}\Omega(1-e^{-t/(40	ext{ }\mu	ext{F})(0.9	ext{ M}\Omega)})$
Simplifying, we get: 
$t=2.37	ext{ s}$

Now we can use the formula for $Q$ again, with this value of $t$: 
$Q=CV$
$Q=40	ext{ }\mu	ext{F}(20	ext{ V})$
$Q=800	ext{ }\mu	ext{C}$

Therefore, the answer is A) 1600 µC, because the charge on the capacitor is double this value when the potential difference across the capacitor is twice the potential difference across the resistor.

Question 15

For the circuit shown in the figure, C = 12 µF and R = 8.5 MΩ. Initially the switch S is open with the capacitor charged to a voltage of 80 V. The switch is then closed at time t = 0.00 s. What is the charge on the capacitor, when the current in the circuit is 3.3 µA?

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Question 16

A 8.0-μF uncharged capacitor is connected in series with a 6.0-kΩ resistor, an ideal 20-V dc source, and an open switch. If the switch is closed at time t = 0.0 s, what is the charge on the capacitor at t = 9.0 ms?

Accepted Answer

Before the switch is closed, the capacitor is uncharged and acts as an open circuit. Once the switch is closed, the capacitor begins to charge through the resistor. The time constant for this circuit is given by:

τ = RC = (6.0 kΩ)(8.0 μF) = 48 ms

After one time constant (t = 48 ms), the capacitor will have charged to approximately 63% of its maximum voltage. Therefore, at t = 9.0 ms, the charge on the capacitor will be:

Q = CV = (8.0 μF)(20 V)(1 - e^(-t/τ)) ≈ (8.0 μF)(20 V)(1 - e^(-0.009/0.048)) ≈ 17% of the maximum charge.

Therefore, the answer is choice C.

Question 17

When an initially uncharged capacitor is charged through a 25-kΩ resistor by a 75-V dc ideal power source, it takes 0.23 ms for the capacitor to reach 50% of its maximum charge? What is the capacitance of this capacitor?

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Quiz 23: Circuits

A 2.0-μF capacitor that is initially uncharged is charged through a 50-kΩ resistor. How long does it take for the capacitor to reach 90% of its full charge?

In the circuit shown in the figure, all the capacitors are initially uncharged when the switch S is suddenly closed, and the battery is ideal. Find (a) the maximum reading of the ammeter and (b) the maximum charge on the 5.00-µF capacitor.

A multiloop circuit is shown in the figure, but some quantities are not labeled. Find the current I2 if the batteries are ideal. (It is not necessary to solve the entire circuit.)

For the circuit shown in the figure, V = 20 V, C = 10 µF, R = 0.80 MΩ, and the battery is ideal. Initially the switch S is open and the capacitor is uncharged. The switch is then closed at time t = 0.00 s. What is the potential difference across the resistor 20 s after closing the switch?

For the circuit shown in the figure, C = 13 µF and R = 7.6 MΩ. Initially the switch S is open with the capacitor charged to a voltage of 80 V. The switch is then closed at time t = 0.00 s. What is the charge on the capacitor 40 s after closing the switch?

A resistor with a resistance of 360 Ω is in a series circuit with a capacitor of capacitance 7.3 × 10-6 F. What capacitance must be placed in parallel with the original capacitance to change the capacitive time constant of the combination to three times its original value?

A circuit contains a 2.0-MΩ resistor in series with an uncharged capacitor. When this combination is connected across an ideal battery, the capacitor reaches 25% of its maximum charge in 1.5 s. What is its capacitance?

A 2.0-μF capacitor is charged to 12 V and then discharged through a 4.0-MΩ resistor. How long will it take for the voltage across the capacitor to drop to 3.0 V?

A fully charged 37-µF capacitor is discharged through a 1.0-kΩ resistor. If the voltage across the capacitor is reduced to 7.6 volts after just 20 ms, what was the original potential across the capacitor?

A series circuit consists of a 2.5-μF capacitor, a 7.6-MΩ resistor, and an ideal 6.0-V dc power source. (a) What is the time constant for charging the capacitor? (b) What is the potential difference across the capacitor 25 s after charging begins?

For the circuit shown in the figure, V = 60 V, C = 20 µF, R = 0.10 MΩ, and the battery is ideal. Initially the switch S is open and the capacitor is uncharged. The switch is then closed at time t = 0.00 s. What is the charge on the capacitor 8.0 s after closing the switch?

For the circuit shown in the figure, C = 12 µF and R = 8.5 MΩ. Initially the switch S is open with the capacitor charged to a voltage of 80 V. The switch is then closed at time t = 0.00 s. What is the charge on the capacitor, when the current in the circuit is 3.3 µA?

A 8.0-μF uncharged capacitor is connected in series with a 6.0-kΩ resistor, an ideal 20-V dc source, and an open switch. If the switch is closed at time t = 0.0 s, what is the charge on the capacitor at t = 9.0 ms?

When an initially uncharged capacitor is charged through a 25-kΩ resistor by a 75-V dc ideal power source, it takes 0.23 ms for the capacitor to reach 50% of its maximum charge? What is the capacitance of this capacitor?

A multiloop circuit is shown in the figure, but some quantities are not labeled. Find the current I₂ if the batteries are ideal. (It is not necessary to solve the entire circuit.)

A resistor with a resistance of 360 Ω is in a series circuit with a capacitor of capacitance 7.3 × 10^-6 F. What capacitance must be placed in parallel with the original capacitance to change the capacitive time constant of the combination to three times its original value?