Capacitance

1. Definitions (Must Know)

A. Capacitor

A capacitor is two conductors separated by an insulator. It stores charge and energy.

B. Capacitance, C

Capacitance, C, is charge stored per unit potential difference:

C = Q/V

Unit: farad (F), where 1 F = 1 C V⁻¹.

Here Q is the magnitude of the charge on either plate. The plates carry equal and opposite charges, + Q and -Q.

2. Key Ideas (What Earns Marks)

• Rearrangements:
  • Q = CV, V = Q/C
• Energy stored in a capacitor is the area under the V–Q graph.
• For a capacitor with constant C, V = Q/C is a straight line through the origin, so the area is a triangle:
  • U = 1/2QV = Q²/2C = 1/2CV²

Data table

V = Q/C (example gradient)
Charge, Q (mC)	Potential difference, V (V)
0	0
1	2
2	4
3	6
4	8
5	10

3. Detailed Explanations

A. What capacitance measures

Capacitance tells you how much charge a capacitor can store for a given potential difference:

• larger C means more charge stored for the same V (since Q = CV).

B. Why energy is the area under the V–Q graph

When you add a small charge dQ to a capacitor at potential V, the work done is:

dW = V dQ

If C is constant, V increases linearly with Q from 0 to V. So the V–Q graph is a straight line and the area under it is a triangle:

U = 1/2QV = 1/2CV²

4. Common Mistakes

• Forgetting to convert muF and nF to farads.
• Using U = QV instead of U = 1/2QV for a capacitor (the voltage rises from 0 to V as it charges).
• Mixing Q in coulombs with “number of electrons” (need Q = Ne if converting).

5. Exam Tips

• If C and V are given: use U = 1/2CV².
• If C and Q are given: use U = Q²/2C.
• If Q and V are given: use U = 1/2QV.
• Quick unit check: F·V² = (C/V)·V² = C·V = J.

6. Worked Examples

Example 1: Find Q and U from C and VCore

A capacitor has C = 220 muF and is charged to V = 12 V. Find the charge stored and the energy stored.

Show Answer

Q = CV = (220 × 10⁻⁶)(12) = 2.64 × 10⁻³ C

U = 1/2CV²; = 1/2(220 × 10⁻⁶)(12²); = 1.58 × 10⁻² J

Example 2: Find C and U from Q and VCore

A capacitor stores Q = 6.0 muC at V = 200 V. Find C and the energy stored.

Show Answer

C = Q/V = 6.0 × 10⁻⁶/200 = 3.0 × 10⁻⁸ F = 30 nF

U = 1/2QV = 1/2(6.0 × 10⁻⁶)(200) = 6.0 × 10⁻⁴ J

Example 3: Use a straight-line V–Q graph (triangle area)Core

The V–Q graph is a straight line through the origin. At Q = 4.0 × 10⁻³ C, V = 8.0 V.

1. Find C.
2. Find the energy stored at that point.

Show Answer

C = Q/V = 4.0 × 10⁻³/8.0 = 5.0 × 10⁻⁴ F

Energy is area under the line (triangle):

U = 1/2QV = 1/2(4.0 × 10⁻³)(8.0) = 1.6 × 10⁻² J

Example 4: Find energy from Q and CCore

A capacitor has capacitance C = 4.0 muF and stores charge Q = 3.0 × 10⁻⁵ C. Find the energy stored.

Show Answer

Use U =; fracQ²2C: U =; frac(3.0; times10⁻⁵)²2(4.0; times10⁻⁶) = 1.13; times10⁻⁴ J

Example 5: Finding voltage from energy and capacitanceCore

A capacitor has C = 2.0 muF and stores energy U = 9.0 × 10⁻⁴ J. Find the voltage across it.

Show Answer

Use U =; tfrac12 CV²: V =; sqrt; frac2UC =; sqrt; frac2(9.0; times10⁻⁴)2.0; times10⁻⁶ =; sqrt9.0; times10² = 30 V

7. Mind Stretchers

Mind stretcher 1: Explain why U = 1/2QV and not U = QVExtension

Show Answer

As the capacitor charges, the potential difference increases from 0 to V.

So the average potential during charging is V/2, and: U = Q(V/2) = 1/2QV

Mind stretcher 2: Unit check for ; tfrac12 CV²Extension

Show that ; tfrac12 CV² has units of joules.

Show Answer

C has unit farad (F) and 1 F = 1 C V⁻¹.

So: CV² = (; mathrmC V⁻¹)V² =; mathrmC V =; mathrmJ since 1 V = 1 J C⁻¹.

Mind stretcher 3: Optional (Enrichment)Extension

A. Calculus form (same idea)

The “area under the V–Q graph” can also be written as:

U = int₀Q V dQ

If C is constant, V = Q/C and this gives U = Q²/2C, which is the same as 1/2QV.

8. Practice, Quiz and Next Step

Close your notes and use Capacitance in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: An unfamiliar data set or physical system requires you to apply Capacitance while stating the model, regime and assumptions.

1. Retrieve: define capacitance in your own words, including units, sign or conditions where relevant.
2. Represent: Choose and label an appropriate diagram, graph, table or symbolic model; derive or justify the relationship used.
3. Apply: Reach a conclusion, then evaluate it using units, uncertainty, a limiting case and one practical or modelling limitation.

Check the response before looking back

• The model, regime, coordinates and assumptions are explicit.
• The derivation or multi-step reasoning is visible rather than implied.
• The conclusion is tested against units, data quality and a limiting case.
• A practical control, uncertainty or model limitation is evaluated where applicable.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theA-Level Physics course hub orpractice browser for an independent re-test.