MHT-CET Physics · Electrostatics

Capacitance and Combinations of Capacitors

Capacitance is the charge stored per volt, C = Q/V, fixed by a capacitor's geometry; in series capacitors share one charge and their reciprocals add, in parallel they share one voltage and they add directly.

All Electrostatics notes MHT-CET Physics strategy

Why this matters

23 PYQs, four HARD: two networks read from a figure, a seven-capacitor puzzle, and two plates with unequal charges. The recurring shapes: C from geometry (plates, a sphere, the Earth), an equivalent capacitance or a 'which arrangement of seven capacitors gives this value' puzzle, how voltage divides in series, and reading C off a V–Q graph.

Concept 1 of 3: Capacitance of Plates and Spheres

A capacitor stores more charge per volt when its plates are bigger and closer: C = ε₀A/d. For a lone sphere the 'other plate' is at infinity and C = 4πε₀R — so even the Earth has a capacitance of under a millifarad.

Definition

$C = \dfrac{Q}{V}$ ; unit farad. Plates at $+20$ V and $-20$ V differ by 40 V.
Parallel plates: $C = \dfrac{\varepsilon_0 A}{d}$ ; circular plates $A = \pi r^2$ , so radius $\times\sqrt{2}$ doubles $A$ .
Field between the plates: $E = \dfrac{Q}{\varepsilon_0 A} = \dfrac{Q}{Cd}$ .
Isolated sphere: $C = 4\pi\varepsilon_0 R$ . For a sphere of volume $V$ and area $A$ , $R = \dfrac{3V}{A}$ .
Plates given unequal charges $q_1, q_2$ : the facing surfaces carry $\pm\dfrac{q_1 - q_2}{2}$ , so $V = \dfrac{q_1 - q_2}{2C}$ .
Charge $Q$ at voltage $V$ , $Q_1$ at $V - V_1$ : $V = \dfrac{QV_1}{Q - Q_1}$ .

Parallel-plate and spherical capacitors

C = \frac{\varepsilon_0 A}{d}, \qquad C_{\text{sphere}} = 4\pi\varepsilon_0 R

Worked example

Plates of area 0.02 m² are 1 mm apart in air. Capacitance?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 22 April Shift II · Q23Moderate

Example 1 · Electrostatics · Capacitance and Combinations of Capacitors

Earth is assumed to be a charged conducting sphere having volume V and surface area A . The capacitance of the earth in free space is (

\varepsilon_{0}=

permittivity of free space)

Half the voltage from ±V plates

Plates at

+20

V and

-20

V have a 40 V difference between them. Using 20 V doubles the capacitance and lands on a printed option.

Drill 5 more on capacitance of plates and spheres

Concept 2 of 3: Series and Parallel Combinations

In series the same charge sits on every capacitor, so the voltages split in inverse proportion to C — the small capacitor takes the big voltage — and the total is less than the smallest. In parallel every capacitor sees the same voltage, so the charges split in proportion to C and the capacitances simply add.

Definition

Series: $\dfrac{1}{C} = \sum\dfrac{1}{C_i}$ , same $Q$ , $V_i \propto \dfrac{1}{C_i}$ . $n$ identical: $\dfrac{C}{n}$ , and the breakdown voltage becomes $nV$ .
Parallel: $C = \sum C_i$ , same $V$ , $Q_i \propto C_i$ .
'Seven identical capacitors' puzzles: $n$ in parallel, then $7 - n$ in series with that block: $\dfrac{1}{C} = \dfrac{1}{nC_0} + \dfrac{7 - n}{C_0}$ .
Networks from a figure: reduce the innermost series or parallel group first, and redraw after each step.

Combinations

\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots, \qquad C_p = C_1 + C_2 + \cdots

Worked example

4\,\mu

F and

12\,\mu

F are in series across 64 V. Charge and voltage on each?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 14th May Shift 1 · Q42Moderate

Example 2 · Electrostatics · Capacitance and Combinations of Capacitors

Which of the following combination of 7 identical capacitors each of

2\,\mu\text{F}

gives a capacitance of

\dfrac{10}{11}\,\mu\text{F}

Giving the bigger voltage to the bigger capacitor

In series

V \propto \frac{1}{C}

3\,\mu

F and

2\,\mu

F across 100 V put 60 V on the

2\,\mu

F. Asked for

V_2 : V_1

, the answer is

3 : 2

, and

2 : 3

is waiting.

Drill 14 more on series and parallel combinations

Concept 3 of 3: Reading Capacitance From a Graph

On a graph of V against Q the slope is 1/C, so the FLATTER line is the bigger capacitor. Flip the axes to Q against V and the slope is C itself, so the steeper line is bigger. Check the axes before comparing slopes.

Definition

V on the y-axis, Q on the x-axis: slope $= \dfrac{1}{C}$ ; smaller slope, larger $C$ .
Q on the y-axis: slope $= C$ .
Same charge: the line lower on a V–Q graph has less voltage, so more capacitance.

Slope of the line

V = \frac{1}{C}\,Q

Worked example

On a V–Q graph two lines have slopes 2 V/μC and 5 V/μC. Their capacitances?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 16th May Shift 2 · Q5Easy

Example 3 · Electrostatics · Capacitance and Combinations of Capacitors

The graph shows the variation of voltage

V

across the plates of two capacitors A and B versus increase in charge

Q

stored in them. Then

Steeper means bigger — on the wrong graph

With V on the y-axis, the steeper line needs more volts for the same charge: it is the SMALLER capacitor.

Drill 1 more on reading capacitance from a graph

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Capacitance of Plates and Spheres
Parallel-plate and spherical capacitors
$C = \frac{\varepsilon_0 A}{d}, \qquad C_{\text{sphere}} = 4\pi\varepsilon_0 R$
Series and Parallel Combinations
Combinations
$\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots, \qquad C_p = C_1 + C_2 + \cdots$
Reading Capacitance From a Graph
Slope of the line
$V = \frac{1}{C}\,Q$

Watch out for (3)

Half the voltage from ±V plates
→ Capacitance of Plates and Spheres
Giving the bigger voltage to the bigger capacitor
→ Series and Parallel Combinations
Steeper means bigger — on the wrong graph
→ Reading Capacitance From a Graph

Test yourself on Electrostatics

20 past MHT-CET questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Electrostatics chapter test Drill all 23 questions

Related notes

Energy Stored in a Capacitor →