Electrostatic Potential and CapacitanceNEET Physics · Class 12 · NCERT Chapter 2

Introduction

Last chapter you learned about electric force and field. This chapter introduces electric potential, the scalar quantity behind voltage, and capacitance, the device that stores charge in a circuit. NEET picks from this chapter heavily: 1 to 2 questions every year.

Repeated favourites: potential of a point charge and dipole, capacitor combinations (series and parallel), parallel plate capacitor with a dielectric, and energy stored. Five formulas in the cheat sheet cover almost everything.

Electric potential and potential difference

Electric potential V at a point is the work done by an external agent in bringing a unit positive charge from infinity to that point against the field, with no change in kinetic energy:

SI unit: volt (V) equals 1 J per C. Potential difference between A and B is $V_B-V_A=-{\int }_A^B\vec{E}\cdot d\vec{l}$.

Potential due to a point charge

Sign matches the sign of q. V at infinity is taken as zero. Note that V falls as 1/r, slower than the field which falls as 1/r².

Potential of a point charge falls as 1/r. Sign matches the sign of the charge. V is a scalar (not a vector).

Potential due to a system of charges

Potential is a scalar, so for many charges just add (with sign):

No vectors needed. This is why potential is often easier to work with than field for complex problems.

Potential due to multiple charges is the algebraic (signed) sum, not a vector sum. Add up the contributions.

Potential due to a dipole

At distance r from the centre of a short dipole, with theta the angle to the dipole axis:

On the axis (theta = 0): $V=kp/r^2$. On the equator (theta = 90°): $V=0$. Notice V falls as 1/r² (faster than a point charge), and the equator is at zero potential everywhere.

Equipotential surfaces

A surface on which V is constant. Properties:

• Field lines are always perpendicular to equipotential surfaces.
• No work is done in moving a charge along an equipotential surface (since ΔV = 0).
• Two equipotential surfaces never intersect.
• For a point charge: equipotentials are concentric spheres around it.
• For a uniform field: equipotentials are parallel planes perpendicular to the field.

Relation between E and V

Electric field is the negative gradient of potential. In one dimension:

Consequences: where V is constant, E equals 0. Where V varies steeply, E is large. The field points from high potential to low potential.

For a point charge: V falls as 1/r, E falls as 1/r². Their relationship is E = -dV/dr (E is the negative slope of V).

Potential energy of a system of charges

Energy stored in bringing the charges together from infinity. For two charges:

For a system, sum over all distinct pairs:

Three charges have 3 pairs; four charges have 6 pairs (n choose 2).

Conductors and electrostatic shielding

• Inside a conductor in electrostatic equilibrium: E = 0.
• Just outside a charged conductor: $E=\sigma /{ϵ}_0$, perpendicular to the surface.
• Whole conductor is at the same potential (one equipotential).
• All charge resides on the surface in equilibrium.
• Electrostatic shielding: a closed conducting cavity has zero E inside, regardless of outside fields. This is why electronics live in metal boxes.

Capacitor and capacitance

A capacitor is two conductors separated by an insulator. When connected to a battery of voltage V, charge Q moves from one plate to the other, giving plus Q on one and minus Q on the other. Capacitance is defined as:

SI unit: farad (F) equals 1 C per V. 1 F is huge in practice; common values are pF (10⁻¹²), nF (10⁻⁹) and µF (10⁻⁶).

Parallel plate capacitor

Two flat conducting plates of area A separated by distance d. Field between them (uniform): $E=\sigma /{ϵ}_0$. Voltage $V=Ed$. So:

Capacitance increases with bigger plates and decreases with larger separation.

Capacitance of a parallel-plate capacitor. Larger area: more capacitance. Bigger separation: less. Higher dielectric constant K: K times more.

Series and parallel combinations

Parallel

Same V across each. Charges add. Effective C is the sum:

Series

Same charge Q on each. Voltages add. Reciprocals add:

Memory hook: for capacitors, the series and parallel rules are SWAPPED compared to resistors. Capacitors in parallel act like a single bigger capacitor; in series, smaller.

Three capacitors combined two ways. Series gives a smaller effective C; parallel gives a larger one.

Energy stored in a capacitor

To charge a capacitor up to voltage V, work must be done against the existing voltage:

The energy is stored in the electric field between the plates. Energy per unit volume:

Energy in a capacitor lives in the electric field between the plates. Three equivalent formulas, pick whichever is most convenient.

Effect of dielectric

Insert a slab of dielectric constant K between the plates. The molecules of the dielectric polarise, partly cancelling the field, so V drops (at fixed Q) and capacitance rises:

Two scenarios when you insert the dielectric:

• Battery still connected (V fixed): Q goes up by K, U goes up by K. The battery supplied extra charge.
• Battery disconnected (Q fixed): V drops by K, U drops by K. Energy went into pulling the dielectric in.

Inserting a dielectric of constant K multiplies capacitance by K. What changes next depends on whether the battery is still connected.

Worked NEET problems

Summary cheat sheet

• V of point charge: $V=kq/r$.
• System: $V=\sum k{q}_i/r_i$.
• Dipole: $V=kp\cos \theta /r^2$.
• E-V relation: $E=-dV/dr$.
• Equipotential: perpendicular to E, no work done along it.
• U of two charges: $U=k{q}_1{q}_2/r$.
• Capacitance: $C=Q/V$.
• Parallel plate: $C_0={ϵ}_{0}A/d$.
• With dielectric: $C=K{C}_0$.
• Parallel: $C=C_1+C_2+\dots$.
• Series: $1/C=1/C_1+1/C_2+\dots$.
• Energy: $U=\frac{1}{2}C{V}^2=\frac{1}{2}QV=Q^2/(2C)$.

Next: try the interactive widgets for parallel-plate capacitors, combinations and dielectrics, or work through the 32 NEET PYQs with full solutions. To time yourself, take the free 10-question mock test.