Task Sets: Capacitors and Dielectrics - OpenStax Physics (High School)

A

Recall / Warm-Up

1.

Which best describes what a capacitor does in a circuit?

A.

It stores separated electric charge and maintains a potential difference between two conductors.

B.

It converts electrical energy to chemical energy for long-term storage.

C.

It stores electric current and releases it when the circuit opens.

D.

It resists changes in current, similar to an inductor.

2.

A parallel-plate capacitor is connected to a 12 V battery. The voltage across the capacitor is:

A.

12 V, equal to the battery voltage.

B.

Less than 12 V, because the capacitor uses up some voltage.

C.

More than 12 V, because the capacitor amplifies voltage.

D.

Zero, because the capacitor blocks DC.

3.

When a dielectric material is inserted between the plates of a charged capacitor (with the plates disconnected from any battery), what happens to the voltage across the plates?

A.

The voltage decreases because the dielectric reduces the electric field.

B.

The voltage increases because the dielectric increases the capacitance.

C.

The voltage stays the same because charge is conserved.

D.

The voltage increases because the dielectric strengthens the field.

B

Fluency Practice

1.

A capacitor stores $Q = 180\ \mu\text{C}$ of charge when connected to a $V = 9\ \text{V}$ source. What is the capacitance in microfarads?

2.

A $47\ \mu\text{F}$ capacitor is connected to a $5.0\ \text{V}$ source. How much charge is stored on the capacitor, in microcoulombs?

3.

A capacitor with $C = 100\ \mu\text{F}$ stores $Q = 500\ \mu\text{C}$ of charge. A second capacitor with $C = 500\ \mu\text{F}$ stores the same charge $Q = 500\ \mu\text{C}$ . Which capacitor has the higher voltage across its plates?

A.

The $100\ \mu\text{F}$ capacitor — it has a voltage of 5 V, five times higher than the $500\ \mu\text{F}$ capacitor.

B.

The $500\ \mu\text{F}$ capacitor — larger capacitance means larger voltage for the same charge.

C.

Both have the same voltage — they store the same charge.

D.

Cannot be determined without knowing the plate area.

4.

A $200\ \mu\text{F}$ capacitor is charged to $V = 300\ \text{V}$ . Calculate the energy stored in the capacitor in joules.

5.

A capacitor stores $U = 0.050\ \text{J}$ of energy at a voltage of $V = 100\ \text{V}$ . What is the capacitance in microfarads?

C

Mixed Practice

D

Word Problems

1.

A parallel-plate capacitor has plates of area $A = 0.05\ \text{m}^2$ and separation $d = 2.0\ \text{mm}$ . It is connected to a $9.0\ \text{V}$ source. Use $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2/(\text{N·m}^2)$ .

Calculate the charge $Q$ stored on the capacitor in nanocoulombs.

2.

A capacitor ( $C = 32\ \mu\text{F}$ ) is charged to $V_0 = 100\ \text{V}$ . The plate separation is then halved (with the capacitor disconnected from the battery).

1.

What is the new capacitance after the plate separation is halved?

2.

The capacitor is disconnected, so the charge $Q$ stays the same. What is the new voltage across the plates?

3.

A $100\ \text{pF}$ capacitor has a dielectric with $\kappa = 3.2$ inserted between its plates. The capacitor is connected to a $12\ \text{V}$ source after insertion.

1.

Calculate the energy stored in the capacitor with the dielectric, in joules. Express your answer in scientific notation (e.g., 2.3e-8).

2.

Calculate the energy stored in the same capacitor without the dielectric at the same voltage ( $V = 12\ \text{V}$ ), in joules. Express your answer in scientific notation (e.g., 7.2e-9).

E

Error Analysis

1.

Student's explanation:

"A capacitor and a battery both store energy and can power a circuit. The main difference is that a capacitor stores electric current inside it, while a battery stores chemical energy. A fully charged capacitor can supply current to a circuit for just as long as a battery can, just more quickly."

A student describes how a capacitor is different from a battery. Read their explanation and find the error.

A.

The student is wrong that both can power a circuit — only batteries can drive current.

B.

The student wrongly says a capacitor stores current. Capacitors store charge (and energy in the electric field), not current. Also, capacitors discharge quickly and cannot sustain current for as long as a battery.

C.

The student is wrong that a battery stores chemical energy — batteries store charge, like capacitors.

D.

The explanation is correct — the only error is in the word 'current' instead of 'voltage.'

2.

Student's reasoning:

"The $50\ \mu\text{F}$ capacitor has five times the capacitance. Since $Q = CV$ , it stores five times the charge: $Q = (50 \times 10^{-6})(12) = 600\ \mu\text{C}$ vs. $Q = (10 \times 10^{-6})(12) = 120\ \mu\text{C}$ ."

"And since larger capacitance means larger voltage, the $50\ \mu\text{F}$ capacitor also has a higher voltage when charged."

A student solves a problem where a $10\ \mu\text{F}$ capacitor at $12\ \text{V}$ is compared with a $50\ \mu\text{F}$ capacitor at $12\ \text{V}$ . Read their reasoning and identify the error.

A.

The student is wrong about $Q = CV$ — the formula should be $Q = C/V$ .

B.

The charge calculation is correct, but the conclusion about voltage is wrong. Both capacitors are at the same voltage (12 V). Larger capacitance does not mean larger voltage — $C$ and $V$ are inversely related when $Q$ is fixed.

C.

The charge calculation is wrong — the two capacitors store the same charge since they are at the same voltage.

D.

Both the charge calculation and the voltage conclusion are correct.

F

Challenge

1.

A $C = 50\ \mu\text{F}$ capacitor is charged to $V_1 = 200\ \text{V}$ . The voltage is then increased to $V_2 = 400\ \text{V}$ .

By what factor does the stored energy increase? (Calculate the ratio $U_2 / U_1$ .)

2.