Electrostatic Potential and Capacitance Class 12 Notes: All Formulas, Key Points & Free PDF

eSaral › JEE › JEE Main ›Electrostatic Potential and Capacitance Class 12 Physics Notes

🚀 Checkout eSaral Courses

1. Introduction

Class 12 Physics Chapter 2 builds directly on Chapter 1 (Electric Charges and Fields). While Chapter 1 tells you where the field exists, this chapter tells you how much energy a charge carries at any point in that field.

Think of electric potential like height in a gravitational field. Just as a ball rolls from higher ground to lower ground, a positive charge moves from higher potential to lower potential. This analogy will anchor every concept in this chapter.

Understanding this chapter is non-negotiable for JEE aspirants. Questions appear in nearly every JEE Main paper, and the clarity you build here directly helps you in Current Electricity, Moving Charges, and Modern Physics.

2. Core Concepts & Theory

2.1 Electric Potential (V)

The electric potential at a point P is the work done by an external force in bringing a unit positive test charge from infinity to P, at constant velocity (no acceleration, no kinetic energy change).

Formula: V = W / q₀

• V = potential at point P (unit: Volt)
• W = work done (unit: Joule)
• q₀ = test charge (unit: Coulomb)
• 1 Volt = 1 Joule per Coulomb (1 V = 1 J/C)

Potential is a scalar quantity. You can add potentials from multiple charges algebraically — no vector components needed. This is its biggest advantage over electric field calculations.

2.2 Electric Potential Due to a Point Charge

The potential at distance r from a point charge q:

V = kq/r = q / (4πε₀r)

• k = 9 × 10⁹ N·m²/C²
• ε₀ = 8.85 × 10⁻¹² F/m
• r = distance from the charge

Key note: V ∝ 1/r, while E ∝ 1/r². Potential falls slower than the field — a frequent JEE distinction question.

2.3 Electric Potential Due to a Dipole

A dipole has charges +q and −q separated by a distance 2a. Dipole moment p = q × 2a (directed from −q to +q).

V = p·cosθ / (4πε₀r²)

• p = dipole moment (unit: C·m)
• θ = angle from the dipole axis
• r = distance from the centre of the dipole

Special cases:

• At θ = 0° (axial point, same side as +q): V is positive maximum
• At θ = 180° (axial point, same side as −q): V is the negative maximum
• At θ = 90° (equatorial point): V = 0 (but E ≠ 0 here — very important!)

2.4 Potential Due to a System of Charges

For n charges, the total potential at a point is the algebraic sum of individual potentials:

V = (1/4πε₀) × Σ (qᵢ/rᵢ)

This follows the superposition principle. Since potential is scalar, just add all values with their signs.

2.5 Potential Difference

The potential difference between two points A and B:

ΔV = V_B − V_A = W_AB / q

• W_AB = work done in moving charge q from A to B
• Positive ΔV means work is done against the field
• Negative ΔV means the field does the work

2.6 Equipotential Surfaces

An equipotential surface is a surface where every point has the same electric potential. No work is done when a charge moves along this surface (since ΔV = 0, W = qΔV = 0).

Properties of Equipotential Surfaces:

1. Always perpendicular to electric field lines
2. The electric field has no component along the surface
3. Closer surfaces = stronger electric field in that region
4. For a point charge: concentric spheres
5. For a uniform field: parallel planes perpendicular to E
6. For a dipole: complex 3D shapes, but symmetric
7. Two equipotential surfaces never intersect

2.7 Relation Between Electric Field and Potential

The electric field is the negative gradient of potential:

E = −dV/dr

In three dimensions:

• E_x = −∂V/∂x
• E_y = −∂V/∂y
• E_z = −∂V/∂z

The negative sign means E points in the direction of decreasing potential (from high V to low V). This is the electrostatics equivalent of "downhill."

2.8 Electrostatic Potential Energy

Potential energy of a charge q at a point where the potential is V:

U = qV

For a uniform field where V = Ed:

U = qEd

For an electric dipole in a uniform external field:

U = −pE cosθ

• Minimum energy (stable equilibrium): θ = 0° (dipole aligned with field)
• Maximum energy (unstable equilibrium): θ = 180° (dipole anti-aligned with field)
• At θ = 90°: U = 0

3. Important Formulas

4. Conductors, Dielectrics & Capacitors

4.1 Properties of Conductors (Electrostatic Equilibrium)

When no charge is moving, a conductor has these properties:

1. Electric field inside = zero (E = 0)
2. The entire conductor is at the same potential (equipotential body)
3. Excess charge resides only on the outer surface
4. Electric field just outside the surface = σ/ε₀ (perpendicular to surface)
5. Field is strongest at sharp points (high curvature = high charge density)
6. Electrostatic shielding: the interior is completely unaffected by external fields

4.2 Dielectrics and Polarisation

A dielectric is an insulator that gets polarised in an electric field. Its molecules form tiny dipoles aligned with the applied field. These dipoles create an opposing internal field, which reduces the net field within the dielectric.

Types of Polarisation:

Polarisation vector P = Total Dipole Moment / Volume (unit: C/m²)

Dielectric constant (κ or εᵣ): ratio of field in vacuum to field in dielectric. κ = E₀/E (always ≥ 1; for vacuum, κ = 1)

4.3 Capacitors and Capacitance

A capacitor stores electrical energy between two conducting plates separated by a dielectric.

Capacitance: C = Q/V (unit: Farad; 1 F = 1 C/V)

Factors that affect the capacitance of a parallel plate capacitor (PPC):

1. Plate area A — larger area → higher C
2. Plate separation d — smaller gap → higher C
3. Dielectric material — higher κ → higher C

Formula for PPC:

C = ε₀εᵣ A/d

4.4 Effect of Inserting a Dielectric

Two scenarios — this is the most tested concept in JEE:

Scenario A: Battery remains connected (V = constant)

Scenario B: Battery disconnected before inserting dielectric (Q = constant)

Expert Tip (Saransh Gupta, AIR-2): The battery question is the #1 source of wrong answers in this chapter. Always check this first before solving any dielectric problem.

4.5 Combination of Capacitors

Series Combination:

1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + ...

• Charge Q is the same on every capacitor
• Voltage divides: V_total = V₁ + V₂ + V₃
• Equivalent C is always less than the smallest individual capacitor
• Used when higher voltage handling is needed

Parallel Combination:

C_total = C₁ + C₂ + C₃ + ...

• Voltage V is the same across every capacitor
• Charge divides: Q_total = Q₁ + Q₂ + Q₃
• Equivalent C is always greater than the largest individual capacitor
• Used when more charge storage is needed

4.6 Energy Stored in a Capacitor

A charged capacitor stores energy in the electric field between its plates.

U = ½CV² = Q²/2C = ½QV

Choose the form based on what is given:

• Given C and V → use ½CV²
• Given Q and C → use Q²/2C
• Given Q and V → use ½QV

Energy density (energy per unit volume of the field):

u = ½ε₀E²

5. Key Points to Remember

1. Electric potential is scalar; electric field is a vector.
2. Potential can be positive, negative, or zero; field magnitude is always positive.
3. At the equatorial point of a dipole: V = 0 but E ≠ 0. Very common exam trap.
4. E = 0 inside a conductor; V = constant (not necessarily zero).
5. Two equipotential surfaces never intersect.
6. Work done to move a charge along an equipotential surface is zero.
7. Work done to move a charge around any closed loop in an electrostatic field is zero (conservative field).
8. When a dielectric is inserted, always ask: Is the battery connected or not?
9. For PPC: C increases if A increases, d decreases, or κ increases.
10. Energy is released (lost as heat) when two charged capacitors are connected together.

6. Important Questions (JEE + CBSE)

Easy (2 marks each):

1. Define electrostatic potential at a point. Write its SI unit.
2. What is an equipotential surface? State two properties.
3. Why is the electric field inside a conductor zero?
4. State the relation between E and V.
5. What is electrostatic shielding? Give one application.

Medium (3–5 marks each):

6. Derive the expression for potential due to an electric dipole at a general point. State special cases.
7. A 200 pF capacitor is charged to 300 V and disconnected. Its charge is transferred to an uncharged 100 pF capacitor. Find the common potential and energy lost.
8. Three capacitors 2 μF, 4 μF, and 6 μF are connected in series across 24 V. Find equivalent capacitance, charge on each, and voltage across each.
9. Explain polarisation of a dielectric. How does inserting a dielectric affect the capacitance of a PPC?
10. Prove E = −dV/dr. Use it to explain why E is zero inside a conductor.

Hard (5 marks each):

11. A parallel plate capacitor is fully charged by a battery. The battery is disconnected, and a dielectric slab (κ = 3) is inserted. Discuss changes in E, V, Q, C, and U with explanation.
12. Find the work done in assembling four identical charges q at the corners of a square of side a.
13. A capacitor charged to V₀ is connected in parallel to an uncharged capacitor of double the capacitance. Find the common voltage and energy lost.
14. Four capacitors, C₁ = 4 μF, C₂ = 8 μF in one branch, and C₃ = 4 μF, C₄ = 8 μF in another, are connected across 100 V in a network. Find the charge and voltage across each.

7. Tips & Tricks for the Exam

Tip 1 — The Battery Rule (most important) Before solving any dielectric problem, write down: "Battery connected → V constant" or "Battery disconnected → Q constant." This one check prevents the most common mistakes.

Tip 2 — Use All Three Energy Formulas U = ½CV² = Q²/2C = ½QV. Pick whichever avoids a calculation step. If C and V are given, use the first. If Q and C are given, use the second.

Tip 3 — Dipole Potential Shortcut For the equatorial point, V = 0 always (cosθ = cos90° = 0). You don't need to calculate anything — just state it.

Tip 4 — Finding E from V in 3D If V is given as a function of coordinates (e.g., V = 3x² + 2y − z), then: E_x = −∂V/∂x, E_y = −∂V/∂y, E_z = −∂V/∂z Then |E| = √(E_x² + E_y² + E_z²)

Tip 5 — Superposition for Potential (not for field) Potential is scalar — add values directly with signs. For electric field, you need to add vectors. Use potential when asked for the field at a high-symmetry point — it's faster.

Tip 6 — Redraw the Circuit For any capacitor network, redraw clearly and label each component before solving. Identify series groups and parallel groups methodically. This alone eliminates 80% of circuit errors.

Tip 7 — Series Capacitor Trap In a series combination, the smallest capacitor holds the largest voltage and reaches dielectric breakdown first. This appears in JEE as a conceptual MCQ.

Tip 8 — Energy Loss When Capacitors Are Joined When two capacitors are connected (charge redistributes), energy is always lost as heat. Energy loss = ½ × C₁C₂/(C₁+C₂) × (V₁−V₂)². Memorise this for direct substitution.

9. Download PDF

Download the complete Electrostatic Potential and Capacitance Class 12 notes as a free PDF — all formulas, solved examples, key points, and important questions in one printable document.

Prepared by Saransh Gupta (IIT Bombay AIR-41) and the eSaral Academic Team.

→ Download Free PDF at esaral.com/notes

🚀 Checkout eSaral Courses