Capacitors and capacitance

What is a capacitor?

The constant-capacitance model assumes fixed geometry and a linear dielectric at fixed operating conditions. Define C by Q = CV, so an uncharged capacitor still has capacitance even though the numerical ratio 0/0 is undefined.

In plain words: put charge on two conductors that face each other and a potential difference appears between them. Double the charge and the potential difference doubles too. Capacitance is the fixed ratio of the two: how much charge the pair holds for each volt.

Capacitance C = Q/V

• The conductors can be charged by connecting them to the two terminals of a battery. Even a single conductor can serve as a capacitor, taking the other conductor to be at infinity (NCERT page 67).
• The field between the conductors is proportional to Q: double the charge and the field doubles at every point, by Coulomb’s law and the superposition principle (page 67).
• V is the work done per unit positive charge in taking a small test charge from conductor 2 to conductor 1 against this field. So V is also proportional to Q, and their ratio is a constant (NCERT Eq. (2.38), page 67):

• C depends only on the geometry of the two conductors (shape, size, separation) and, as NCERT shows later, on the insulator (dielectric) between them (pages 67–68; see effect of a dielectric).
• For a large C, V is small for a given Q: a capacitor with a large capacitance holds a large charge at a relatively small potential difference (page 68).

Unit: the farad

• The SI unit of capacitance is the farad: 1 F = 1 C V⁻¹ (NCERT page 68). Since 1 V = 1 N C⁻¹ m, this is also 1 C² N⁻¹ m⁻¹ (page 69).
• The farad is a very big unit. The common units are its sub-multiples: 1 µF = 10⁻⁶ F, 1 nF = 10⁻⁹ F and 1 pF = 10⁻¹² F (page 68).
• A fixed capacitor is drawn as two short parallel lines, like the plates; a variable capacitor is the same symbol with an arrow drawn across it (page 68).
• Besides storing charge, a capacitor is a key element of most ac circuits, taken up in NCERT Chapter 7 (page 68).

Dielectric strength

• A high potential difference means a strong field around the conductors. A strong field can ionise the surrounding air and drive the ions to the oppositely charged plates, partly neutralising them: the charge leaks away as the medium loses its insulating power (NCERT page 68).
• The dielectric strength of a medium is the largest field it can withstand without breaking down as an insulator. For air a common approximate value is 3 × 10⁶ V m⁻¹; actual breakdown depends on pressure, gap and electrode geometry. For conductors about 1 cm apart, this field corresponds to a potential difference of about 3 × 10⁴ V (page 68).
• So, to store a large charge without leaking, the capacitance must be high enough that the potential difference, and with it the field, stays below the breakdown limit. There is a limit to the charge a given capacitor can store without significant leaking (page 68).

Solved examples

Common mistakes to avoid

Quick revision summary

• A capacitor is two conductors separated by an insulator, carrying Q and −Q with a potential difference V; its total charge is zero.
• The field and V are both proportional to Q, so C = Q/V is a constant, set by the conductors’ shape, size and separation and the insulator between them.
• The SI unit is the farad, 1 F = 1 C V^⁻¹ = 1 C^² N^⁻¹ m^⁻¹, a very large unit; µF, nF and pF are used in practice.
• Dielectric strength is the largest field an insulator can take without breaking down; for air it is about 3 × 10^⁶ V m^⁻¹, which limits the charge a capacitor can hold.
• An isolated sphere is a capacitor with C = 4πε_₀R.

Questions students ask

What are the SI unit and dimensional formula of capacitance?

The SI unit of capacitance is the farad, 1 F = 1 C V^⁻¹ (NCERT page 68), and its dimensional formula is [M^⁻¹ L^⁻² T^⁴ A^²]: from C = Q/V, charge [A T] divided by potential [M L^² T^⁻³ A^⁻¹] (see dimensional formulae; beyond the NCERT text, NIST Special Publication 811 gives the farad as m^⁻² kg^⁻¹ s^⁴ A^²). The table on NCERT page 78 prints T^⁻⁴ for capacitance, a misprint; its entry for potential is right. The same units give ε₀ in F m^⁻¹: from C = ε₀A/d, ε₀ = Cd/A = 8.854 × 10^⁻¹² F m^⁻¹ (reasoned from NCERT Eq. (2.43), page 69).

What is the difference between a capacitor and capacitance?

A capacitor is a device, two conductors separated by an insulator that carry charges Q and −Q, while capacitance is a property of it, the charge it holds per volt of potential difference between the conductors, C = Q/V, measured in farads (NCERT pages 67–68). Every capacitor has a capacitance fixed by its shape, size, spacing and dielectric, but capacitance does not need a manufactured capacitor: a single conductor has one too, with the other conductor taken at infinity (NCERT page 67), and so does any pair of conductors near each other.

Why does a capacitor use two conductors close together instead of one?

A capacitor uses two conductors close together because the nearby −Q lowers the potential difference needed to hold a given +Q, which raises C = Q/V: NCERT notes that a capacitor confines the field lines to a small region, so even a strong field gives only a small potential difference between the conductors (NCERT Points to Ponder 2, page 78). Beyond the NCERT text, compare an isolated metal sphere of radius 5.0 cm, C = 4πε₀R ≈ 5.6 pF, with two plates of 0.010 m^² facing each other 0.10 mm apart, C = ε₀A/d ≈ 890 pF. The plates hold about 160 times as much charge per volt, though their area is smaller than the sphere’s.

What is the difference between dielectric strength and dielectric constant?

Dielectric strength is the largest electric field an insulator can withstand before it breaks down, about 3 × 10^⁶ V m^⁻¹, or 3 kV mm^⁻¹, for air (NCERT page 68), while the dielectric constant K is a pure number, the factor by which the medium filling the gap raises a capacitor’s capacitance (NCERT page 70). So dielectric strength limits the voltage a capacitor can take, about strength × gap for parallel plates, while K multiplies the charge it holds per volt. A good capacitor dielectric needs both: a high K and a high strength, so that a thin gap does not break down.

What is the difference between a capacitor and a battery?

A capacitor stores energy by keeping equal and opposite charges apart on two conductors, the energy sitting in the electric field between them, while a cell or battery stores chemical energy and uses it to keep a steady current flowing in a circuit (NCERT pages 73–75; NCERT Chapter 3, pages 93–94; see emf of a cell). As a capacitor discharges, its potential difference falls in step with its charge, V = Q/C, all the way to zero, whereas a cell’s emf is set by its electrodes and electrolyte. A capacitor returns the electric-field energy stored during charging; it does not replenish that energy through a chemical reaction.

What is the difference between a farad and a faraday?

The farad, symbol F, is the SI unit of capacitance: 1 F = 1 C V^⁻¹ (NCERT page 68). One faraday is an amount of charge, approximately 96 485 C, the magnitude carried by one mole of electrons. Distinguish this from the Faraday constant, F = N_Ae, whose unit is C mol^⁻¹: the exact SI values of N_A and e give approximately 96 485.33 C mol^⁻¹ (NIST SI defining constants). NCERT Chemistry uses about 96 500 C mol^⁻¹ in calculations (Class 12, Part I, Unit 2, page 52).

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A 2.0 µF capacitor is connected across 50 V. What is the charge on it?

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   Q = CV = 2.0 × 10⁻⁶ × 50 = 1.0 × 10⁻⁴ C (NCERT Eq. (2.38), page 67).

2. Q2. Express 1 F in terms of coulombs, newtons and metres.

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   1 F = 1 C V⁻¹ = 1 C (N C⁻¹ m)⁻¹ = 1 C² N⁻¹ m⁻¹ (NCERT page 69).

3. Q3. Why does the charge on a capacitor leak away if the potential difference is too high?

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   The strong field ionises the surrounding air, and the ions are driven to the oppositely charged plates, partly neutralising them (NCERT page 68).