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3.6 Energy Stored by a Capacitor

Unlike a resistor, a capacitor stores energy rather than dissipates it. Recalling the equation of the capacitor action, we define the energy storage process as:

$p = v \cdot i = C \cdot v \frac{d v}{d t}$ (3.14)

$w = \int p d t = C \int v \frac{d v}{d t} d t = C \int_{- \infty}^{t} v d v = \frac{1}{2} C \cdot v^{2}$ (3.15)

The process of energy storage in a capacitor follows a linear process (Fig. 3.16).

A line graph depicting the relationship between charge (Q) on the y-axis and voltage (V) on the x-axis. The line has a positive slope, indicating that the gradient represents capacitance. The area under the line is labeled as the work done, measured in joules.

Fig. 3.16 Energy stored in a capacitor. Source Own elaboration

练习题

A capacitor with capacitance $C = 100 μ F$ is charged to a voltage of $V = 50 V$ . How much energy is stored in the capacitor?

0.125 J

0.25 J

2.5 J

5.0 J

What is the fundamental difference between a resistor and a capacitor in terms of energy handling?

A. Both components store energy equally well.

B. A resistor stores energy while a capacitor dissipates it.

C. A capacitor stores energy while a resistor dissipates it.

D. Neither component interacts with electrical energy.

The power relation for a capacitor is given by $p = C \cdot v \frac{d v}{d t}$ . If the voltage across a $10 μ F$ capacitor is increasing at a rate of $1000 V/s$ when $v = 20 V$ , what is the instantaneous power being stored?

0.2 W

0.4 W

2.0 W

200 W

Which of the following statements about energy storage in a capacitor are correct?

A. The energy storage process in a capacitor follows a linear process.

B. A capacitor dissipates energy as heat like a resistor.

C. The energy stored depends on both the capacitance and the square of the voltage.

D. The work done to charge a capacitor equals the area under the charge-voltage curve.

In the charge-voltage relationship graph for a capacitor, the gradient (slope) of the line represents the capacitance value.

The energy stored in a capacitor is directly proportional to the voltage across it (i.e., $w \propto v$ ).

The work done in charging a capacitor is represented by the area under the charge-voltage curve and is measured in joules.

The formula for the energy stored in a capacitor is $w =$ ___ $C \cdot v^{2}$ .

Using the relationship $Q = C V$ and the fact that the energy stored equals the area under the charge-voltage curve, explain why the energy formula contains a factor of $\frac{1}{2}$ .

A capacitor stores energy as voltage by accumulating charge between its plates. If the voltage across a capacitor doubles while the capacitance remains constant, how does the stored energy change? Explain your reasoning.

A capacitor with capacitance $C = 100 μ F$ is charged to a voltage of $V = 50 V$ . Using the relationship $Q = C V$ and the energy formula $w = \frac{1}{2} C V^{2}$ , what is the energy stored in the capacitor and the charge stored on its plates?

A. Energy =

0.125 J

, Charge =

5 mC

B. Energy =

0.125 J

, Charge =

2 mC

C. Energy =

0.250 J

, Charge =

5 mC

D. Energy =

0.250 J

, Charge =

2 mC

Which of the following statements correctly describe the relationship between the charge-voltage characteristic and energy storage in a capacitor? Select all that apply.

A. The gradient of the Q-V line represents capacitance.

B. The area under the Q-V line represents the work done in charging the capacitor.

C. The energy stored can be expressed as

w = \frac{1}{2} Q V

D. The Q-V relationship is linear for an ideal capacitor.

E. The energy stored is proportional to the square of the voltage.

In an R-C circuit, when a capacitor is fully charged, the potential difference across the capacitor equals that of the source, and the energy stored is given by $w = \frac{1}{2} C V^{2}$ . If the capacitor has $C = 200 μ F$ and the source voltage is $U_{0} = 20 V$ , the maximum energy stored is ___ J.

When a capacitor discharges through a resistor in an R-C circuit, the energy stored in the capacitor decreases as the voltage decreases according to $w = \frac{1}{2} C V^{2}$ , where the voltage follows the discharge equation $u = E \cdot e^{- t / τ}$ .

Explain why a capacitor stores energy rather than dissipates it like a resistor, and describe what happens to this stored energy when the capacitor discharges through a resistor.

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3.6 Energy Stored by a Capacitor