Revision: Electrostatics >> Electrostatic Potential and Capacitance Physics Science (English Medium) Class 12 CBSE

Electrostatic potential energy is the energy stored in a system of charges due to their relative positions.

Core Idea: If an external force moves a test charge against the electrostatic force, the work done by the external agent is stored as electrostatic potential energy.

Electrostatic potential at a point is the work done by an external agent in bringing a unit positive test charge slowly from infinity to that point without acceleration.

The potential difference between two points P and R is the work done by an external force in moving a unit positive test charge from one point to the other.

The work done by an external agent in bringing a unit positive test charge slowly from infinity to a point in an electric field, against the electrostatic force, is called the electric potential at that point.

An electric dipole is a pair of equal and opposite charges separated by a small distance.

If the charges are separated by a distance 2a2a, then 2a2a is called the dipole length.

The electrostatic potential V at a point in an electric field is defined as the work done by an external force in bringing a unit positive charge (without acceleration) from infinity to that point.

The surface at which electric potential is the same at each point is called an equipotential surface.

The electrostatic force is a conservative force — the work done in moving a charge between two points is independent of the path and depends only on the initial and final positions. This is why the potential energy is well-defined.

The electrostatic potential energy of a system of charges is defined as the work done by an external agent in assembling the charges at their respective positions, bringing each charge from infinity, without any kinetic energy being imparted.

• Symbol: U
• SI Unit: Joule (J)
• Nature: Scalar quantity
• Reference: U = 0 when all charges are at infinity

The potential energy of a charge q placed at a point with external electric potential V(r) is equal to the work done in bringing the charge from infinity to that point against the external field.

The condition in which charges in a conductor are at rest, and no further motion of charges occurs.

A conductor in electrostatic equilibrium is an equipotential body, meaning all points on it are at the same electric potential.

The phenomenon in which the electric field inside a cavity of a conductor is zero, irrespective of external charges or fields, is called electrostatic shielding.

Surface charge density is the charge per unit area on the surface of a conductor and is denoted by \[\sigma\].

A polar molecule is a molecule that has a permanent dipole moment even in the absence of an external electric field.

Examples: HCl, H2O.

A dielectric is an insulating material in which free charge carriers are absent or negligible.

An electric dipole consists of two equal and opposite charges separated by a small distance.

A non-polar molecule is a molecule whose centres of positive and negative charge coincide, so its net dipole moment is zero in the normal state.

Examples: O2, H2.

Polarisation is the electric dipole moment per unit volume of a dielectric.

Electric susceptibility \[\chi_e\] is a property of a dielectric that measures how easily it gets polarised in an external electric field.

The ratio of the charge Q given to one of the conductors of a capacitor to the potential difference V between the conductors is called its capacitance, given by C = Q/V.

The ability of a conductor to store charge is called the capacity of conductor.

The maximum electric field that a dielectric medium can withstand without breakdown (of its insulating property) is called its dielectric strength.

A system consisting of two conductors having equal and opposite charges separated by an insulator or dielectric is called a capacitor.

A capacitor that consists of two large, parallel, conducting plates separated by a small distance is called a parallel plate capacitor.

The ratio of the permittivity of a medium to the permittivity of vacuum.

K = ε / ε₀

OR

Dielectric constant is the factor by which the capacitance of a capacitor increases when a dielectric is completely inserted between its plates.

The product of vacuum permittivity and dielectric constant of the medium.

ε = ε₀K

A dielectric is a non-conducting (insulating) material in which charges are bound to their atoms/molecules and cannot move freely. When placed in an external electric field, the molecules of the dielectric get polarised — they develop induced dipole moments that partially oppose the external field.

The process by which the molecules of a dielectric develop induced dipole moments when placed in an external electric field. The induced dipole moments align opposite to the field, creating an opposing induced field E_P.

The maximum electric field a dielectric can withstand before it breaks down (becomes conducting). Measured in V/m. Example: Air ≈ 3 × 10⁶ V/m.

The work done per unit charge in moving a charge from one plate of a capacitor to the other is called the potential difference between the plates.

The capacitance of a single capacitor that stores the same charge at the same voltage as the entire combination is called the equivalent capacitance of the combination.

If the potential energies at points P and R are U_P and U_R​, then

\[V_P-V_R=\frac{U_P-U_R}{q}\]

If the work done in bringing charge q from infinity to point P is W, then

V_P ​= \[\frac {W​}{q}\]

V = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q}{r}\]

Varies on spherical shell carrying charge q and radius R:

• Inside shell (r < R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{R}\]
• On surface (r = R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{R}\]
• Outside shell (r > R): V = \[\frac {1}{4πε_0}\] ⋅ \[\frac {q}{r}\]

\[V=\frac{Q}{4\pi\varepsilon_0r}\]

Potential due to System of Charges:

\[U=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}\]

U = \[\frac{1}{4\pi\varepsilon_0}\cdot\frac{q_1q_2}{r_{12}}\]

\[V(r)=\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\]

• V(r) = electric potential at distance rr from the charge
• q = source charge
• ε₀ = permittivity of free space
• K = dielectric constant of medium
• Reference is taken such that V(∞) = 0.

\[V=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{p\cos\theta}{r^{2}}=\frac{1}{4\pi\varepsilon_{0}}\cdot\frac{\vec{p}\cdot\vec{r}}{r^{3}}(r>>a)\]

The electric dipole moment is:

\[\vec{p}=q(2a)\hat{p}\]

Its direction is from the negative charge to the positive charge.

V = \[\frac{W_{\infty\to P}}{q_{0}}\] (Work done per unit positive charge)

SI Unit: Volt (V) = Joule/Coulomb (J/C);
Dimensional Formula: [M¹L²T⁻³A⁻¹]

When a charge q₀​ is moved from point A to point B on the same equipotential surface:

Since V_A = V_B​ on the surface:

\[V=\frac{1}{4\pi\varepsilon_{0}}\left[\frac{q_{1}}{r_{1}}+\frac{q_{2}}{r_{2}}+\frac{q_{2}}{r_{3}}+\frac{q_{4}}{r_{4}}+.........+\frac{q_{n}}{r_{n}}\right]\]

\[V=\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^{i=n}\frac{q_i}{r_i}\]

U(r) = qV(r)

where:

• U(r) = Potential energy of the charge at position r (in Joules, J)
• q = Charge of the particle (in Coulombs, C)
• V(r) = External electric potential at position r (in Volts, V)
• r = Position vector of the point from the origin

For a dipole making an angle θ with a uniform electric field:

τ = pE sin θ

In vector form: τ = p × E

This torque rotates the dipole toward the field direction.

\[\vec{E}=\frac{\sigma}{\varepsilon_0}\hat{n}\]

where

• σ = surface charge density
• \[\hat n\] = outward normal unit vector
• \[\varepsilon_0\] = permittivity of free space.

Magnitude form:

E = \[\frac{\sigma}{\varepsilon_0}\]

Vector form:

$$\vec{E} = \frac{\sigma}{\varepsilon_0}\hat{n}$$

For a linear isotropic dielectric, polarisation is directly proportional to the electric field:

C = \[\frac {2πkε₀ l}{2.303 log(b/a)}\]

C = Q/V

C = 4πkε₀ · [\[\frac {ab}{(b − a)}\]]

For two plates separated by distance d:

\[C=\frac{\varepsilon_0A}{d}\]

With a dielectric medium:

\[C=\frac{K\varepsilon_0A}{d}\]

\[{C_P=C_1+C_2+C_3+\cdots}\]

For n identical capacitors of capacitance C each: C_P = nC

Physical Insight: Adding capacitors in parallel is like adding more storage tanks — the total storage capacity simply increases.

For two capacitors in series, the voltage across each is:

\[V_1=\frac{C_2}{C_1+C_2}\cdot V\]

\[V_2=\frac{C_1}{C_1+C_2}\cdot V\]

Physical Insight: The smaller the capacitor, the larger the voltage drop across it in a series combination. This is why identical series capacitors share voltage equally.

\[{\frac{1}{C_S}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots}\]

For n identical capacitors of capacitance C each: C_S = \[\frac {C}{n}\]

W = \[\frac {1}{2}\]qV

OR

U = \[\frac {Q^2}{2C}\] ​= \[\frac {1​}{2}\]QV = \[\frac {1}{2}\]​CV²

SI unit: Joule (J)

• Electric potential at a point is the work done per unit positive test charge in bringing it slowly from infinity to that point, against the electric field.
• For a point charge q in air/vacuum:
  V(r) = \[\frac{1}{4\pi\varepsilon_0}\frac{q}{r}\]
• In a medium of dielectric constant K:
  V(r) = \[\frac{1}{4\pi\varepsilon_0K}\frac{q}{r}\]
• Positive charge produces positive potential; negative charge produces negative potential.
• Potential due to a point charge is spherically symmetric and depends only on distance r.
• Distance dependence:
  F ∝ 1/r², E ∝ 1/r², V ∝ 1/r.
• The potential at infinity is taken as zero; only potential differences are physically significant.
• The electrostatic field is conservative, so the work done in moving a charge between two points is path independent.

Capacitors in Series:

Equivalent capacitance: \[\frac{1}{C_s}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots\]

• Same voltage (V) across all capacitors
• Charge divides
• The equivalent capacitance is greater than the largest capacitor

Capacitors in Parallel:

\[C_p=C_1+C_2+C_3+\cdots\]

• Same voltage (V) across all capacitors
• Charge divides
• The equivalent capacitance is greater than the largest capacitor