Electric Potential and Capacitance: Complete Chapter Revision for Class 12 by PW

Electric Potential and Capacitance is an important Class 12 Physics chapter that connects electric field, work, energy, and charge configurations. The chapter covers electric potential energy, potential difference, equipotential surfaces, conductors, dipoles, and the behaviour of charges in electric fields.

PW Electric Potential and Capacitance Class 12 notes offer important formulas, relations, examples and comparison tables. The chapter also covers capacitors, dielectric effects, energy stored in capacitors and capacitor networks, helping you revise the concepts and numerical applications together.

Electric Potential Energy

Electric potential energy represents the work associated with a particular configuration of charges. 

• For a conservative force: Change in U = − Work Done by Conservative Force

• For an external agent: Work Done by External Agent = Change in U

Taking potential energy at infinity as zero, the potential energy of a charge q at point P is the work done in bringing it slowly from infinity to P.

For two point charges q₁ and q₂ separated by distance r: U = kq₁q₂/r

Potential energy is a scalar quantity and may be positive or negative depending on the signs of the charges.

For a pair of charges, use the algebraic signs of both charges directly in U = kq₁q₂/r.

Potential Energy of a System of Charges

The potential energy of a system of charges is obtained by adding the energy associated with every distinct pair of charges. For n charges, the number of distinct pairs is:

N = n(n − 1)/2

For three equal charges q placed at the corners of an equilateral triangle of side a: U = 3kq²/a

For four equal charges at the corners of a square of side l: U = 4(kq²/l) + 2(kq²/√2l)

Therefore: U = (4 + √2)kq²/l

The energy of the complete system should not be confused with the energy associated with only one charge or one pair. A system containing five charges has 10 distinct pair terms.

Work Done During Charge Rearrangement

When a charge configuration changes from an initial state to a final state:

External Work = Uᶠ − Uⁱ

Work Done by Electric Field = Uⁱ − Uᶠ

For example, if three equal charges at the corners of an equilateral triangle are moved so that the side changes from a to 2a:

Uᵢ = 3kq²/a

Uᶠ = 3kq²/2a

Therefore: External Work = −3kq²/2a

The negative sign indicates that the electric field does positive work during the rearrangement.

Electric Potential And Potential Difference

Electric potential is the electric potential energy per unit charge: V = U/q

For a point charge Q: V = kQ/r

Electric potential is a scalar quantity, so potential contributions from different charges are added algebraically.

The important relations are:

• U = qV

• ΔU = qΔV

• External Work = qΔV

• Work Done by Electric Field = −qΔV

Potential is measured in volt (V), where: 1 V = 1 J/C

For example, if a charge moves from a point at 10 V to another point at 30 V: ΔV = 30 − 10 = 20 V

Hence, the external work is 20q.

Potential Gradient And Electric Field

Electric field and electric potential are related by: E = −dV/dr

The quantity dV/dr is called the potential gradient. The negative sign indicates that electric potential decreases in the direction of the electric field.

• In vector form: dV = −E · dr

• For a potential-versus-distance graph: Slope = −E

• For an electric-field-versus-distance graph: ΔV = − Area under the E-r graph

Moving perpendicular to the electric field does not change the potential. Therefore, all points on an equipotential surface have the same potential.

Superposition Of Potential And Electric Field

Electric potential follows the principle of superposition, but since potential is scalar, its contributions are added algebraically: Vₙₑₜ = V₁ + V₂ + V₃ + ...

Electric-field contributions are added vectorially because electric field has both magnitude and direction.

For two charges of the same sign, their net potential cannot be zero at a finite point because their potential contributions have the same sign. However, the electric field can be zero if the field vectors cancel each other.

Therefore, depending on the charge arrangement, it is possible to have:

• Zero electric field and non-zero potential

• Zero potential and non-zero electric field

• Both electric field and potential non-zero

• Both electric field and potential zero

At points equidistant from a point charge, the potential is the same, while the electric-field vectors may have different directions.

Potential Energy And Motion

When a charge moves in the direction of the electric force, its potential energy decreases while its kinetic energy increases, provided no other work changes the energy.

For like charges moving apart, potential energy decreases. Similarly, for unlike charges moving closer, potential energy decreases.

For a repelled charge moving from infinity towards a fixed charge, the closest approach can be obtained using conservation of energy: 1/2 mv² = kQq/r₀

Therefore: r₀ = 2kQq/mv²

Thus, the closest approach is inversely proportional to the square of the initial speed. If the initial speed is doubled, the closest approach becomes one-fourth.

Electric Potential Due to Rings, Spheres and Dipoles

The potential due to a charged ring or arc can be calculated using: V = kQ/R

when every charge element is at the same distance R from the point of observation.

At the centre of a charged ring, the potential is non-zero while the electric field is zero because the electric-field contributions cancel vectorially.

• For a point on the axis of a ring of radius R: V = kQ/√(R² + x²)

• At large distances, the ring behaves approximately like a point charge, giving: V ≈ kQ/x

• For a uniformly charged hollow spherical shell: Vᵢₙₛᵢdₑ = kQ/R

• and outside the shell: Vₒᵤₜₛᵢdₑ = kQ/r

• A conducting solid sphere has the same potential behaviour as a conducting hollow shell.

• For a uniformly charged non-conducting solid sphere: Vᵢₙₛᵢdₑ = kQ(3R² − r²)/(2R³)

• At the centre: V = 3kQ/2R

• For an ideal electric dipole: V = k p cosθ/r²

On the equatorial plane, θ = 90°, so the potential is zero. On the axial line, the magnitude of potential is maximum for the same distance.

Conductors in Electrostatic Equilibrium

A conductor contains free charge carriers that can move through the material. In electrostatic equilibrium:

• The conductor is at constant potential.

• The net electric field inside the conductor is zero.

• The electric field just outside the surface is perpendicular to the surface.

• The surface electric field is related to surface charge density by E = σ/ε₀ in vacuum.

Charge density is generally greater near sharp regions of a conductor, resulting in a stronger electric field there. However, the potential remains the same throughout the conductor in electrostatic equilibrium.

When a conducting sphere is placed in an external electric field, its free charges redistribute themselves and produce an induced field. At equilibrium:

External Field + Induced Field = 0

Hence, the net electric field inside the conductor remains zero.

Capacitor and Capacitance

A capacitor is a device used to store electric charge and electrostatic energy. Capacitance is defined as: C = Q/V

Its SI unit is the farad (F).

• For an isolated spherical conductor of radius r: C = 4πε₀r

• For two conductors carrying equal and opposite charges: C = Q/|V₁ − V₂|

• For a parallel-plate capacitor having plate area A and separation d: C = ε₀A/d

• When a dielectric of dielectric constant κ is inserted between the plates: C = κε₀A/d

Thus, inserting a dielectric increases the capacitance by a factor of κ, under the corresponding fixed-geometry conditions.

Energy Stored in a Capacitor

The energy stored in a capacitor can be expressed in three equivalent forms:

• U = 1/2 CV²

• U = Q²/2C

• U = 1/2 QV

The energy density between the plates is: u = ε₀E²/2

It can also be written as: u = σ²/(2ε₀)

The electrostatic pressure between capacitor plates has the same expression as the energy density for the ideal parallel-plate case.

The relationship between energy and capacitance depends on what remains fixed:

Capacitor Networks and Short Circuits

A short circuit connects two points at the same potential. Such points are treated as a single node while simplifying a capacitor network.

The general steps are:

1. Identify all points connected by continuous wire.

2. Assign the same node to these points.

3. Remove a capacitor connected between points belonging to the same node.

4. Combine the remaining capacitors in series or parallel.

The physical appearance of a circuit is less important than its electrical connectivity.

For four identical capacitors C connected in series: Cₑq = C/4

For three identical capacitors in parallel, connected in series with another capacitor C: Cₚ = 3C

Therefore: Cₑq = (3C × C)/(3C + C) = 3C/4

Electric Potential and Capacitance connects the concepts of work, electric field, potential energy, potential difference, conductors and capacitors. These PW Electric Potential and Capacitance Class 12 notes bring together important formulas, relations, capacitor combinations, dielectric effects and electric-field concepts in one place. Revising the sign conventions, graph-based relations, conductor properties and capacitor formulas can help you recall the chapter more efficiently while preparing for Class 12 Physics questions and numerical problems.