Electrostatic Potential and Capacitance
स्थिरवैद्युत विभव तथा धारिता | Class 12 Physics - Chapter 2 | NCERT + RBSE 2026
| Subject: | Physics |
| Class: | 12 (RBSE + NCERT) |
| Chapter: | 2 |
| Topic: | Electrostatic Potential and Capacitance |
| Year: | 2025-26 |
| Status: | Complete & Verified |
• Coulomb constant: k = 9 × 10⁹ N·m²/C²
• Permittivity of free space: ε₀ = 8.854 × 10⁻¹² C²/(N·m²)
• Potential unit: 1 V = 1 J/C (Volt = Joule per Coulomb)
• Capacitance unit: 1 F = 1 C/V (Farad = Coulomb per Volt)
Learning Objectives
After completing this chapter, students will be able to:
- Define electrostatic potential and potential difference between two points
- Calculate potential due to point charges, dipoles, and systems of charges
- Understand equipotential surfaces and their relationship with electric field
- Calculate potential energy of charge systems in various configurations
- Apply the six fundamental properties of conductors in electrostatic equilibrium
- Explain dielectric polarisation and its effect on electric field
- Define capacitance and analyze parallel plate capacitors
- Solve problems involving series and parallel combinations of capacitors
- Calculate energy stored in capacitors using multiple equivalent formulas
1. Introduction
Chapter 1: Electric Field (Vector quantity)
• Direction matters
• Requires vector addition
↓
Chapter 2: Electric Potential (Scalar quantity)
• Only magnitude
• Algebraic addition
• Simplifies many calculations
↓
Applications: Capacitors, Energy Storage, Circuits
In Chapter 1, we studied electric field and the forces experienced by charges. Now we introduce electrostatic potential, a scalar quantity that provides an alternative and often simpler way to analyze electric phenomena.
This chapter also introduces capacitors, devices that store electric charge and energy, which are fundamental components in electronic circuits, energy storage systems, and modern technology.
2. Electrostatic Potential
Infinity (V = 0) ←————— Test charge q₀ ————→ Point P
Work done by external force = Wext
Potential at P: V = Wext/q₀
2.1 Definition
Electrostatic potential at a point is defined as the work done per unit positive charge in bringing a test charge from infinity to that point against the electric field.
SI Unit: Volt (V) = Joule/Coulomb (J/C)
Nature: Scalar quantity (can be positive, negative, or zero)
- Potential is a scalar quantity (only magnitude, no direction)
- Zero of potential is chosen at infinity (V∞ = 0)
- Potential can be positive (due to +ve charge) or negative (due to −ve charge)
- Potential difference is independent of the path taken
2.2 Potential Difference
Potential difference between two points A and B is the work done per unit charge in moving a test charge from A to B.
If VA > VB, positive charge moves naturally from A to B
2.3 Relation Between Electric Field and Potential
Or in integral form:
Physical Meaning: Electric field points in the direction of maximum decrease of potential. The negative sign indicates that field points from higher to lower potential.
3. Potential Due to a Point Charge
Charge Q at origin
Point P at distance r
If Q > 0: Potential is positive
If Q < 0: Potential is negative
At r → ∞: V → 0
where:
- k = 9 × 10⁹ N·m²/C²
- Q = charge (can be +ve or −ve)
- r = distance from charge
- ε₀ = 8.854 × 10⁻¹² C²/(N·m²)
- Positive charge (Q > 0): V > 0 (positive potential)
- Negative charge (Q < 0): V < 0 (negative potential)
- At infinity: V = 0 (reference point)
- Comparison with field: E = kQ/r² (note: V ∝ 1/r, E ∝ 1/r²)
4. Potential Due to an Electric Dipole
Charges: −q ←—— 2a ——→ +q
Dipole moment: p = q(2a)
Point P at distance r, angle θ
Axial (θ = 0°): V = kp/r²
Equatorial (θ = 90°): V = 0
For a dipole consisting of charges +q and −q separated by distance 2a:
where:
- p = q(2a) = dipole moment
- θ = angle from dipole axis
- r = distance from dipole center
- On axial line (θ = 0° or 180°): V = ±kp/r²
- On equatorial plane (θ = 90°): V = 0 (equal contributions from +q and −q cancel)
- Note: Potential falls off as 1/r² (faster than point charge)
5. Potential Due to a System of Charges
Multiple charges: q₁, q₂, q₃, ...
at distances: r₁, r₂, r₃, ...
Total potential = SCALAR SUM
V = V₁ + V₂ + V₃ + ...
(No vector addition needed!)
For a system of multiple point charges, the total potential at any point is the algebraic sum of potentials due to individual charges.
Key Advantage: Unlike electric field (vector), potentials add algebraically (scalar addition)!
6. Equipotential Surfaces
Point Charge: Concentric spheres
Uniform Field: Parallel planes perpendicular to field
Dipole: Complex 3D surfaces
Key: All points on surface have same V
Equipotential surface is a surface on which electric potential has the same value at all points.
- No work is done in moving a charge along an equipotential surface (W = q∆V = 0)
- Electric field is always perpendicular to equipotential surface at every point
- Equipotential surfaces never intersect each other
- For a point charge: equipotential surfaces are concentric spheres
- For uniform electric field: equipotential surfaces are parallel planes perpendicular to field
- Equipotential surfaces are closer where field is stronger
6.1 Relation Between Field and Potential
Or equivalently:
The negative sign indicates that electric field points in the direction of decreasing potential (from higher V to lower V).
7. Potential Energy of a System of Charges
Bring q₁ from ∞: Work = 0
Bring q₂ from ∞: Work = kq₁q₂/r₁₂
Bring q₃ from ∞: Work = kq₃(q₁/r₁₃ + q₂/r₂₃)
Total PE = Sum of all pair interactions
7.1 Potential Energy of Two Charges
where r is the separation between charges q₁ and q₂
Sign: U > 0 (repulsive), U < 0 (attractive)
7.2 Potential Energy of Multiple Charges
Sum is taken over all distinct pairs (i < j)
8. Potential Energy in an External Field
1. Self Energy: Interaction between charges in system
2. External Energy: Interaction with external field
Total PE = Self PE + External PE
8.1 Potential Energy of a Single Charge
where V is the potential at the location of charge q due to external sources
8.2 System of Two Charges in External Field
Total potential energy includes both self-energy and interaction with external field:
where V₁ and V₂ are potentials at locations of q₁ and q₂ due to external sources
8.3 Potential Energy of a Dipole in External Field
Vector form: U = −p⃗ · E⃗
- θ = 0° (aligned with field): U = −pE (minimum, stable equilibrium)
- θ = 90° (perpendicular): U = 0
- θ = 180° (opposite to field): U = +pE (maximum, unstable equilibrium)
- Work to rotate: W = ∆U = pE(cos θ₁ − cos θ₂)
9. Electrostatics of Conductors
Free electrons redistribute instantly
↓
Result: Six Fundamental Properties
1. E = 0 inside
2. Field ⊥ surface
3. Charge on surface only
4. V = constant
5. E = σ/ε₀ at surface
6. Interior shielded
Conductors have six important properties in electrostatic equilibrium:
9.1 Inside a Conductor, Electrostatic Field is Zero
In electrostatic equilibrium, the electric field inside a conductor is zero. Free electrons redistribute themselves on the surface to cancel any internal field.
If E ≠ 0 inside, charges would move, contradicting the assumption of electrostatic equilibrium.
9.2 At the Surface, Field Must Be Normal
At the surface of a charged conductor, the electrostatic field must be perpendicular (normal) to the surface at every point.
If there were a tangential component of E, surface charges would experience a force and would move along the surface, which contradicts equilibrium.
9.3 The Interior Can Have No Excess Charge
In static situation, all excess charge on a conductor resides entirely on its surface. The interior has no net charge.
9.4 Electrostatic Potential is Constant
The electrostatic potential is constant throughout the volume of the conductor and has the same value on its surface as inside.
Reason: Since E = 0 inside, no work is done in moving a charge, hence V is constant
9.5 Electric Field at the Surface
where σ is the surface charge density
Note: This is twice the field of an isolated charged sheet (which is σ/2ε₀)
9.6 Electrostatic Shielding
The interior of a hollow conductor is shielded from external electric fields. External charges induce charges on the outer surface, but the field inside remains zero.
- Faraday Cage: Metal enclosure protects interior from external fields
- Coaxial Cables: Outer conductor shields signal from interference
- Car Protection: Metal body shields passengers during lightning
- Sensitive Equipment: Enclosed in metal boxes in laboratories
10. Dielectrics and Polarisation
Without Dielectric: E₀, C₀
↓ Insert dielectric
With Dielectric: E = E₀/K, C = KC₀
Field decreases, Capacitance increases
(K = dielectric constant)
Dielectric: An insulating material that, when placed between capacitor plates, increases capacitance.
10.1 Polarisation
When a dielectric is placed in an external electric field, its molecules develop or align dipole moments.
- Polar molecules (e.g., H₂O): Have permanent dipole moments that align with external field
- Non-polar molecules (e.g., N₂): Develop induced dipole moments in external field
Result: Net alignment of dipoles creates an internal field opposing the external field
10.2 Dielectric Constant
Ratio of field without dielectric to field with dielectric
| Material | Dielectric Constant (K) |
|---|---|
| Vacuum | 1 (exactly) |
| Air | 1.00054 ≈ 1 |
| Paper | 3.7 |
| Glass | 5-10 |
| Mica | 6 |
| Water | 80 |
- K ≥ 1 for all materials (K = 1 for vacuum only)
- Higher K means greater polarisation and greater capacitance increase
- Dielectrics reduce electric field: E = E₀/K
11. Capacitors and Capacitance
Two conductors (plates)
Separated by insulator
Apply voltage V → Stores charge Q
Capacitance: C = Q/V
Capacitor: A device consisting of two conductors carrying equal and opposite charges, used to store electric charge and energy.
11.1 Definition of Capacitance
SI Unit: Farad (F) = Coulomb/Volt (C/V)
- Capacitance is a measure of ability to store charge
- 1 Farad is a very large unit (typical capacitors are μF, nF, pF)
- 1 μF = 10⁻⁶ F, 1 nF = 10⁻⁹ F, 1 pF = 10⁻¹² F
- Capacitance depends only on geometry and dielectric, not on Q or V
12. The Parallel Plate Capacitor
Two parallel plates
Area: A
Separation: d
Charge: +Q on one, −Q on other
Field between plates: E = σ/ε₀
Voltage: V = Ed
Capacitance: C = ε₀A/d
The most common type of capacitor consists of two parallel conducting plates of area A separated by distance d.
where:
- A = area of each plate (m²)
- d = separation between plates (m)
- ε₀ = 8.854 × 10⁻¹² F/m
- C ∝ A: Larger area → more capacitance
- C ∝ 1/d: Smaller separation → more capacitance
- Capacitance is independent of charge Q or voltage V
13. Effect of Dielectric on Capacitance
Before: C₀ = ε₀A/d
↓ Insert dielectric (constant K)
After: C = Kε₀A/d = KC₀
Capacitance increases by factor K
When a dielectric material of dielectric constant K is inserted between the plates:
where C₀ is capacitance without dielectric
Dielectric reduces the electric field between plates (E = E₀/K), which reduces the potential difference (V = Ed) for the same charge. Since C = Q/V, capacitance increases.
14. Combination of Capacitors
SERIES: C₁ —— C₂ —— C₃
• Same Q on all
• Voltages add
• 1/Ceq = 1/C₁ + 1/C₂ + 1/C₃
PARALLEL: C₁ ∥ C₂ ∥ C₃
• Same V on all
• Charges add
• Ceq = C₁ + C₂ + C₃
14.1 Capacitors in Series
For two capacitors:
- Same charge Q on all capacitors
- Voltages add: V = V₁ + V₂ + V₃
- Equivalent capacitance is less than the smallest individual capacitance
- Used to increase voltage rating
14.2 Capacitors in Parallel
- Same voltage V across all capacitors
- Charges add: Q = Q₁ + Q₂ + Q₃
- Equivalent capacitance is greater than the largest individual capacitance
- Used to increase charge/energy storage capacity
| Property | Series | Parallel |
|---|---|---|
| Formula | 1/Ceq = Σ(1/Ci) | Ceq = ΣCi |
| Charge (Q) | Same on all capacitors | Divides: Q = Q₁ + Q₂ |
| Voltage (V) | Divides: V = V₁ + V₂ | Same on all capacitors |
| Ceq | Less than smallest | Greater than largest |
| Application | High voltage rating | High capacitance |
15. Energy Stored in a Capacitor
Work done in charging capacitor
= Energy stored in electric field
U = (1/2)QV = (1/2)CV² = (1/2)Q²/C
(All three forms equivalent)
Choose form based on given quantities
- U = (1/2)QV: When both Q and V are given
- U = (1/2)CV²: When C and V are given (most common)
- U = (1/2)Q²/C: When Q and C are given
- All three give the same answer (they're equivalent)
15.1 Energy Density in Electric Field
Unit: J/m³ (Joule per cubic meter)
Energy per unit volume stored in electric field
Practice Questions
Multiple Choice Questions (20 MCQs)
Q1. SI unit of electric potential is:
(a) Joule
(b) Volt
(c) Coulomb
(d) Newton
1 Volt = 1 Joule/Coulomb (J/C)
Q2. Potential due to a point charge Q at distance r is:
(a) kQ/r
(b) kQ/r²
(c) kQ/r³
(d) kQr
V = kQ/r. Compare with electric field E = kQ/r²
Q3. On the equatorial plane of a dipole, potential is:
(a) Maximum
(b) Minimum
(c) Zero
(d) Infinite
At θ = 90°, V = (kp cos 90°)/r² = 0
Q4. Work done in moving a charge on an equipotential surface is:
(a) Maximum
(b) Zero
(c) Infinite
(d) Negative
W = q∆V = 0 (since ∆V = 0 on equipotential surface)
Q5. Electric field lines and equipotential surfaces are:
(a) Parallel
(b) Perpendicular
(c) At 45°
(d) Can be at any angle
Field lines are always perpendicular to equipotential surfaces
Q6. Electric field inside a conductor in electrostatic equilibrium is:
(a) Maximum
(b) Zero
(c) Uniform
(d) Variable
First fundamental property of conductors
Q7. Excess charge on a conductor resides:
(a) Throughout volume
(b) Only on surface
(c) At center
(d) Uniformly distributed
Third property: No excess charge inside conductor
Q8. Potential inside a charged conductor is:
(a) Zero
(b) Constant
(c) Variable
(d) Infinite
Fourth property: Vinside = Vsurface = constant
Q9. Electric field just outside a conductor surface is:
(a) σ/2ε₀
(b) σ/ε₀
(c) 2σ/ε₀
(d) Zero
Fifth property: E = σ/ε₀ (perpendicular to surface)
Q10. Dielectric constant K for all materials is:
(a) K < 1
(b) K = 1
(c) K ≥ 1
(d) K ≤ 1
K = 1 only for vacuum, K > 1 for all other materials
Q11. SI unit of capacitance is:
(a) Volt
(b) Coulomb
(c) Farad
(d) Joule
1 Farad = 1 Coulomb/Volt (C/V)
Q12. For parallel plate capacitor, C = :
(a) ε₀A/d
(b) ε₀d/A
(c) ε₀Ad
(d) A/ε₀d
C ∝ A (area) and C ∝ 1/d (separation)
Q13. When dielectric is inserted in a capacitor, capacitance:
(a) Decreases
(b) Increases
(c) Remains same
(d) Becomes zero
C = KC₀ where K > 1
Q14. In series combination of capacitors:
(a) Q same, V different
(b) V same, Q different
(c) Both same
(d) Both different
Charge is same on all, voltages add: V = V₁ + V₂
Q15. Equivalent capacitance in series is:
(a) Greater than largest
(b) Less than smallest
(c) Average of all
(d) Zero
1/Ceq = 1/C₁ + 1/C₂ + ... makes Ceq < smallest C
Q16. In parallel combination, Ceq = :
(a) 1/C₁ + 1/C₂
(b) C₁ + C₂
(c) C₁C₂
(d) C₁/C₂
In parallel: Ceq = C₁ + C₂ + C₃ + ...
Q17. Energy stored in capacitor U = :
(a) QV
(b) (1/2)QV
(c) 2QV
(d) Q/V
Also U = (1/2)CV² = (1/2)Q²/C (all equivalent)
Q18. Relation between E and V is:
(a) E = dV/dr
(b) E = −dV/dr
(c) E = V/r
(d) E = −V/r
Negative gradient: field points from high to low potential
Q19. Potential energy of dipole in uniform field:
(a) pE cos θ
(b) −pE cos θ
(c) pE sin θ
(d) −pE sin θ
Minimum (−pE) when aligned (θ = 0°)
Q20. Energy density in electric field u = :
(a) ε₀E²
(b) (1/2)ε₀E²
(c) (1/2)ε₀E
(d) ε₀E
Energy per unit volume in electric field
Formula Sheet
| Concept | Formula | Unit |
|---|---|---|
| Potential | V = W/q | V (Volt) |
| Point Charge | V = kQ/r | V |
| Dipole | V = (kp cos θ)/r² | V |
| Potential Difference | VAB = VA − VB | V |
| E and V relation | E = −dV/dr | V/m |
| PE (two charges) | U = kq₁q₂/r | J |
| PE (dipole) | U = −pE cos θ | J |
| Field at surface | E = σ/ε₀ | N/C |
| Dielectric constant | K = E₀/E | Dimensionless |
| Capacitance | C = Q/V | F (Farad) |
| Parallel plate | C = ε₀A/d | F |
| With dielectric | C = KC₀ | F |
| Series | 1/Ceq = Σ(1/Ci) | F |
| Parallel | Ceq = ΣCi | F |
| Energy (Form 1) | U = (1/2)QV | J |
| Energy (Form 2) | U = (1/2)CV² | J |
| Energy (Form 3) | U = (1/2)Q²/C | J |
| Energy density | u = (1/2)ε₀E² | J/m³ |
Previous Chapter: Chapter 1 - Electric Charges and Fields (Coulomb's law, Gauss's law, electric field)
Current Chapter: Chapter 2 - Electrostatic Potential and Capacitance (Potential, energy, capacitors)
Next Chapter: Chapter 3 - Current Electricity (Ohm's law, circuits, resistance)
Understanding potential and capacitance builds directly on electric field concepts from Chapter 1 and provides the foundation for analyzing circuits in Chapter 3.
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