Electrostatic Potential and Capacitance

Concise notes • Key formulas • 50 MCQs with answers

Chapter overview

This chapter builds intuition about how charges create potential and how electric energy is stored and managed in systems. You will connect electric field to potential, compute work and potential energy for point charges and continuous charge distributions, and understand capacitors, dielectrics, and combinations in series/parallel. Mastery here powers many topics: circuits, energy storage, electrostatics in materials, and real-world devices from camera flashes to DRAM.

Potential (V): Work done per unit charge to bring a test charge from infinity to a point. Zero level is a reference; differences matter most.
Relation to field: E is the spatial rate of change of V; uniform fields give linear potential variation.
Capacitance (C): Ability to store charge at a given potential difference; geometry and dielectric govern C.
Dielectrics: Polarizable media reduce internal field, raising capacitance by factor κ (relative permittivity).

Key formulas

Potential and field

V = k Q / r, where k = 1/(4πϵ₀)
ΔV = - ∫ E · dl; for uniform field: ΔV = -E d cosθ
E = -∇V (in 1D: E = - dV/dx)
Potential of multiple point charges: V = Σ k Qᵢ / rᵢ
Potential energy (two charges): U = k Q₁Q₂ / r
Potential energy (N charges): U = ½ Σᵢ Qᵢ V(rᵢ)

Capacitance and energy

C = Q / V
Parallel-plate: C = ϵ₀ A / d (vacuum); with dielectric: C = κ ϵ₀ A / d
Energy in a capacitor: U = ½ C V² = ½ Q V = Q² / (2C)
Series: 1/C_eq = Σ 1/Cᵢ; Parallel: C_eq = Σ Cᵢ
Field in dielectric: E = E₀ / κ (ideal, linear, uniform slab)

Core concepts

Electrostatic potential and zero reference intuition

Only potential differences are physical; choose V = 0 at infinity for isolated charges. Conductors in electrostatic equilibrium are equipotential: the entire conductor is at one potential, so E = 0 inside and field is normal at the surface.

Work and energy in assembling charges applications

Energy stored reflects work required to build the configuration from infinity. For continuous distributions, integrate k dq/r. Energy density in an electric field: u = ½ ϵ E² (in linear dielectric or vacuum).

Capacitors, series/parallel, and dielectrics exam focus

Series reduces equivalent capacitance and shares the same charge; parallel increases it and shares the same voltage. Dielectrics increase capacitance by κ and reduce the field; bound charges create polarization effects.

Edge effects and idealizations realism

Parallel-plate formulas assume negligible fringing, uniform separation, and linear dielectric; practical designs account for edge fields, breakdown strength, and temperature coefficients.

50 MCQs with highlighted answers

Each question has one correct answer, highlighted in green.

1) SI unit of electrostatic potential is:
  • Joule
  • Volt
  • Newton
  • Farad
2) Potential at a distance r from an isolated point charge Q (in vacuum) is:
  • V = (1/4πϵ₀) · Q/r
  • V = (1/4πϵ₀) · Q/r²
  • V = (1/4πϵ₀) · r/Q
  • V = (1/4πϵ₀) · Qr
3) Relation between electric field and potential is:
  • E = ∇V
  • E = -∇V
  • V = ∇E
  • V = -∇E
4) The potential due to multiple point charges equals:
  • Product of individual potentials
  • Vector sum of potentials
  • Algebraic sum of scalar potentials
  • Zero if charges are equal and opposite
5) Work done by the electric field in moving a charge between two points equals:
  • q(V_final + V_initial)
  • q(V_initial − V_final)
  • q(V_final − V_initial) always positive
  • Independent of potential difference
6) In electrostatic equilibrium, the electric field inside a conductor is:
  • Uniform and non-zero
  • Zero at the center only
  • Zero everywhere inside
  • Parallel to the surface
7) The surface of a conductor in electrostatic equilibrium is:
  • At varying potential
  • Potential depends on position
  • An equipotential surface
  • Potential equals zero only
8) The capacitance C is defined by:
  • C = V/Q
  • CV = Q²
  • C = Q/V
  • C = V²/Q
9) SI unit of capacitance is:
  • Volt
  • Ohm
  • Joule
  • Farad
10) For a parallel-plate capacitor (vacuum) with plate area A and separation d:
  • C = ϵ₀ d / A
  • C = ϵ₀ A / d
  • C = A / (ϵ₀ d)
  • C = d / (ϵ₀ A)
11) With dielectric of relative permittivity κ inserted fully between plates, capacitance becomes:
  • C/κ
  • κC
  • C unchanged
  • κ² C
12) Energy stored in a capacitor carrying charge Q and potential V is:
  • U = QV
  • U = ½ QV
  • U = Q²V
  • U = ½ V/Q
13) Equivalent capacitance for capacitors in series satisfies:
  • 1/C_eq = Σ (1/Cᵢ)
  • C_eq = Σ Cᵢ
  • C_eq = Π Cᵢ
  • C_eq = max(Cᵢ)
14) In parallel combination of capacitors:
  • Same charge on each
  • Same potential across each
  • Same energy in each
  • Same charge density
15) In series combination of capacitors:
  • Same potential across each
  • Same charge on each
  • Same capacitance for each
  • Same energy for each
16) The energy density of an electric field in vacuum is:
  • u = ϵ₀/E²
  • u = ½ ϵ₀ E²
  • u = ½ E/ϵ₀
  • u = ϵ₀² E
17) For a uniform electric field E, the potential difference between points separated by distance d along field is:
  • ΔV = -E d
  • ΔV = +E d
  • ΔV = -E/d
  • ΔV = 0
18) Equipotential surfaces are always:
  • Parallel to electric field lines
  • Perpendicular to electric field lines
  • Coincident with field lines
  • Randomly oriented
19) Potential at the center of a uniformly charged ring (total charge Q, radius R) is:
  • 0
  • V = (1/4πϵ₀) · Q/R
  • V = (1/4πϵ₀) · Q/R²
  • V = (1/4πϵ₀) · R/Q
20) Potential at the center of a uniformly charged solid sphere (total charge Q, radius R) is:
  • V = (1/4πϵ₀) · (3Q/2R)
  • V = (1/4πϵ₀) · (Q/2R)
  • V = 0
  • V = (1/4πϵ₀) · (2Q/R)
21) A conductor cavity enclosing no charge has electric field inside the cavity equal to:
  • Non-zero uniform
  • Zero
  • Depends on external field
  • Infinite
22) When a dielectric slab completely fills a capacitor while disconnected from battery, charge on plates:
  • Increases
  • Remains same
  • Decreases
  • Becomes zero
23) In the same situation (dielectric inserted, battery disconnected), potential difference:
  • Increases
  • Decreases by factor κ
  • Unchanged
  • Becomes zero
24) If the battery remains connected while inserting dielectric, the charge on plates:
  • Increases by factor κ
  • Decreases
  • Unchanged
  • Becomes zero
25) Dimensions of ϵ₀ (permittivity of free space) are:
  • [M⁻¹ L⁻³ T⁴ A²]
  • [M L T⁻²]
  • [M L² T⁻²]
  • [M⁰ L⁰ T⁰]
26) Potential difference between plates of a charged parallel-plate capacitor with uniform field E and separation d is:
  • V = Ed²
  • V = E d
  • V = E/d
  • V = E² d
27) For two capacitors C and 2C in series across V, potential across C is:
  • (2/3) V
  • (1/3) V
  • V/2
  • V
28) For two capacitors C and 2C in parallel across V, charge on 2C is:
  • Q = 2C · V
  • Q = C · V
  • Q = V/2C
  • Q = V/C
29) Electric field just outside a charged conductor surface (surface charge density σ) is:
  • E = σ/ϵ₀ (tangent)
  • E = σ/ϵ₀ (normal outward)
  • E = 0
  • E = 2σ/ϵ₀ (tangent)
30) Potential difference between two points on the same equipotential surface is:
  • Positive
  • Negative
  • Zero
  • Depends on path
31) If V = ax² (a > 0), the electric field along x is:
  • E = -2ax (along −x for x>0)
  • E = +2ax
  • E = ax
  • E = -ax²
32) A capacitor is charged to V and disconnected; if plate separation is doubled (vacuum), final potential becomes:
  • V/2
  • 2V
  • V
  • 4V
33) For a spherical capacitor with inner radius a and outer radius b (vacuum), capacitance is:
  • C = 4πϵ₀ ab
  • C = 4πϵ₀ ab/(b − a)
  • C = 4πϵ₀ (b − a)/ab
  • C = 4πϵ₀ (a + b)
34) For a cylindrical capacitor (coaxial) per unit length with radii a < b, capacitance per length is:
  • C′ = 2πϵ₀ / ln(b/a)
  • C′ = 2πϵ₀ ln(b/a)
  • C′ = ϵ₀ ln(a/b)
  • C′ = 2π/ϵ₀ ln(b/a)
35) A dielectric reduces the internal electric field primarily because of:
  • Increase in free charges
  • Polarization and bound surface charges
  • Temperature rise
  • Magnetic effects
36) Potential due to an electric dipole (p) at a point on axial line at distance r (r ≫ size):
  • V = (1/4πϵ₀) · p/r²
  • V = (1/4πϵ₀) · p/r³
  • V = (1/4πϵ₀) · pr
  • V = (1/4πϵ₀) · p/r
37) Potential due to a dipole at a point on the equatorial line (r ≫ size):
  • V = (1/4πϵ₀) · p/r²
  • V = 0
  • V = (1/4πϵ₀) · 2p/r²
  • V = (1/4πϵ₀) · p/r
38) For V = kQ/r, as r → ∞, the potential tends to:
  • kQ
  • 0
  • −∞
39) If a metal sphere is given some charge, it resides:
  • Uniformly throughout the volume
  • Only on the outer surface
  • Only at the center
  • On inner and outer surfaces equally
40) Capacitance depends on:
  • Geometry and dielectric medium
  • Potential alone
  • Charge alone
  • Mass of plates
41) The potential at the surface of a charged conductor is highest at:
  • Flatter regions
  • Sharper (smaller radius) regions for field; potential is uniform over the conductor
  • All points different
  • Depends on environment only
42) Capacitors in series have equivalent capacitance:
  • Less than the smallest individual C
  • Between smallest and largest C
  • Greater than largest C
  • Equal to sum of all C
43) Capacitors in parallel have equivalent capacitance:
  • Less than the smallest C
  • Between smallest and largest C
  • Equal to sum of all C
  • Equal to harmonic mean
44) If Q is kept constant and C is doubled, the stored energy:
  • Quadruples
  • Doubles
  • Halves
  • Unchanged
45) If V is kept constant and C is doubled, the stored energy:
  • Halves
  • Doubles
  • Unchanged
  • Reduces to one-fourth
46) The capacitance of an isolated conducting sphere of radius R (vacuum) is:
  • C = 4πϵ₀ R
  • C = 4πϵ₀ / R
  • C = ϵ₀ / (4πR)
  • C = R / (4πϵ₀)
47) Which is true for an ideal capacitor?
  • Stores charge with energy loss
  • Blocks DC (after charging) and allows AC (reactively)
  • Converts electric energy to heat
  • Provides a constant current source
48) The SI unit of 1/(4πϵ₀) is closest to:
  • F·m
  • N·m²/C²
  • V·m/C
  • J/C
49) For a given capacitor, breakdown occurs when:
  • Q exceeds a certain value regardless of V
  • Electric field exceeds the dielectric strength
  • Temperature is absolute zero
  • Plates are too wide
50) Which change increases capacitance of a parallel-plate capacitor the most?
  • Increase plate separation
  • Decrease plate area
  • Insert higher-κ dielectric fully
  • Reduce ϵ₀

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