NEET Electrostatic Potential and Capacitance Notes by PW

Preparing for Electrostatic Potential and Capacitance requires more than memorising formulas. You need to understand the relationship between electric field and potential, calculate potential energy for systems of charges, interpret graphs, and apply the correct conditions when solving capacitor problems. Regular revision of important formulas, sign conventions, and charge combinations can help improve concept clarity and accuracy while solving numerical and conceptual questions.

Instead of revising individual concepts separately, you can use Physics Wallah’s Electrostatic Potential and Capacitance notes to revise important formulas, concepts, and their applications. These notes help you review electric potential, potential energy, capacitance, capacitor combinations, and related numerical problems in one place.

Work–Energy Theorem and Conservative Forces

The work–energy theorem states:

Net work done by all forces = Change in kinetic energy

This theorem applies to the net work done by all forces acting on an object.

A conservative force does path-independent work and has an associated potential energy. Gravitational, spring and electric forces are conservative, while friction is generally non-conservative.

For a conservative force:

Work done by a conservative force = − Change in potential energy

W = −ΔU

For an external agent making a quasistatic change against a conservative force:

Work done by the external agent = Change in potential energy

Wₑₓₜ = ΔU

If the work done by non-conservative forces is zero, mechanical energy remains conserved:

Change in (kinetic energy + potential energy) = 0

Kᵢ + Uᵢ = Kf + Uf

Conservation of mechanical energy is a special case of the work–energy theorem.

Choosing the Correct Relation

• Work done by a conservative force: W = −ΔU

• Work done by an external agent in a quasistatic change: Wₑₓₜ = ΔU

• Speed after release or projection: Apply conservation of mechanical energy where appropriate.

• Direct work questions: Identify the force whose work is being calculated before choosing the relation.

Electrostatic Potential Energy of Charges

For two point charges q₁ and q₂ separated by a distance r:

U = kq₁q₂/r

Here, k = 1/(4πε₀).

The algebraic signs of both charges must be used. Potential energy is generally taken to be zero when the charges are infinitely far apart:

U(∞) = 0

Electrostatic potential energy is a scalar quantity. Its positive or negative sign indicates the energy relative to the chosen reference, not a direction.

Potential Energy of a System of Charges

For a system of n charges:

U = Σᵢ₍₍ⱼ₎ kqᵢqⱼ/rᵢⱼ

The sum includes every distinct pair of charges exactly once.

The number of distinct pairs is:

Number of pairs = n(n − 1)/2

For example, three charges form three pairs, while four charges form six pairs.

Memory tip: For n charges, the number of distinct pairs is n(n − 1)/2.

When calculating the energy associated with one particular charge, count only the pairs involving that charge. Do not include interactions among the remaining charges.

Electric Potential and Electric Field

Electric potential is potential energy per unit charge:

V = U/q

For a point charge Q:

V = kQ/r

Potential is a scalar quantity, so potentials due to multiple charges are added algebraically:

Vₜₒₜₐₗ = Σᵢ kqᵢ/rᵢ

Electric potential is measured in volts:

1 V = 1 J/C

The electric field and potential are related by:

Eₓ = −dV/dx

In vector form:

E = −∇V

The integral relation is:

Vᵦ − Vₐ = −∫ₐᵦ E · dl

Thus, potential decreases when moving in the direction of the electric field and increases when moving opposite to it. Moving perpendicular to the electric field produces no potential change along that displacement.

Potential and Electric Field Graphs

• Potential function V: Differentiate to find the electric field.

• V–x graph: The slope gives −Eₓ.

• Electric field E: Integrate to find the potential difference.

• Eₓ–x graph: The negative signed area gives the change in potential.

Important: Do not integrate potential to find the electric field. Differentiate the potential function. If the electric field is given, integrate it to calculate the potential difference.

If potential is constant throughout a region:

E = 0

However, zero potential at one point does not necessarily mean that the electric field is zero there. The electric field depends on the spatial variation of potential.

Potential Due to Special Charge Distributions

Uniformly Charged Ring

For a uniformly charged ring of radius R and total charge Q, the potential at its centre is:

V = kQ/R

At a point on its axis, at a distance x from the centre:

V = kQ/√(R² + x²)

For a uniformly charged semicircular ring with linear charge density λ:

Q = λπR

Therefore, the potential at the centre is:

V = kλπ

Spherical Conducting Shell

For a spherical conducting shell of radius R:

• Outside the shell, r > R:
  V = kQ/r

• On the surface, r = R:
  V = kQ/R

• Inside the shell, r < R:
  V = kQ/R

The potential is constant inside a conducting shell.

Uniformly Charged Solid Sphere

For a uniformly charged solid sphere:

• Outside the sphere:
  V = kQ/r

• At the surface:
  V = kQ/R

• At the centre:
  V = 3kQ/(2R)

Potential depends on the reference point, but potential difference does not.

Electric Dipole Potential

An electric dipole consists of equal and opposite charges separated by a distance 2a. Its dipole moment is:

p = q(2a) = 2qa

At a point on the equatorial line, the potentials due to the two charges cancel:

Vₑqᵤₐₜₒᵣᵢₐₗ = 0

At a general point sufficiently far from a short dipole:

V = kp cos θ/r²

The dipole potential is zero on the equatorial line, but the electric field there is not zero. The equatorial electric field is directed opposite to the dipole moment.

Equipotential Surfaces and Conductors

An equipotential surface has the same potential at every point.

Important properties include:

• Work done along an equipotential surface is zero.

• The electric field is perpendicular to an equipotential surface.

• A conductor in electrostatic equilibrium is an equipotential body.

• The electric field inside a conductor is zero.

• Immediately outside a charged conductor:
  E = |σ|/ε₀

At a sharp point, the radius of curvature is small. Hence, the surface charge density and electric field can be greater at sharp regions.

For two conducting spheres connected by a wire, their potentials become equal. If their radii are r₁ and r₂:

q₁/r₁ = q₂/r₂

Therefore:

q₁/q₂ = r₁/r₂

The smaller sphere has greater surface charge density and a greater surface electric field under the usual approximation that the spheres are sufficiently far apart.

Capacitance and Capacitor Basics

Capacitance is defined as:

C = Q/V

For an isolated spherical conductor:

C = 4πε₀R

A parallel-plate capacitor has capacitance:

C = ε₀A/d

With a dielectric of dielectric constant K completely filling the space between the plates:

C = Kε₀A/d

Thus, capacitance increases with plate area and decreases with plate separation.

Series and Parallel Capacitors

Series combination

For capacitors connected in series:

• Charge magnitude is the same on each capacitor.

• Potential difference divides among the capacitors.

• Equivalent capacitance is less than the smallest individual capacitance.

The equivalent capacitance is:

1/Cₑq = Σᵢ 1/Cᵢ

For N identical capacitors:

Cₑq = C/N

Parallel combination

For capacitors connected in parallel:

• Potential difference is the same across each capacitor.

• Charge divides according to capacitance.

• Equivalent capacitance is the sum of the individual capacitances.

The equivalent capacitance is:

Cₑq = Σᵢ Cᵢ

For N identical capacitors:

Cₑq = NC

Memory tip: Series means the same charge magnitude; parallel means the same potential difference.

Charging and Energy Stored in a Capacitor

When an initially uncharged capacitor is connected to a battery:

• At the instant of connection, an ideal capacitor behaves like a short circuit.

• After complete charging in a DC circuit, it behaves like an open circuit.

• The final potential difference equals the battery voltage.

• The final charge is Q = CV.

The energy stored in a capacitor is:

U = ½CV²

Equivalent forms are:

U = Q²/(2C)

U = ½QV

For an initially uncharged capacitor charged to voltage V by an ideal constant-voltage battery through a resistor, the battery supplies energy CV², but only half is stored in the capacitor. The remaining half is dissipated during charging.

If the battery remains connected, the voltage stays constant. If the battery is disconnected, the charge stays constant, provided there is no leakage or conducting path.

Dielectrics in Capacitors

For dielectric layers placed along the separation direction, the arrangement behaves like capacitors connected in series:

Cₑq = ε₀A/(d₁/K₁ + d₂/K₂ + …)

For dielectric regions placed side by side, the arrangement behaves like capacitors connected in parallel:

Cₑq = ε₀(K₁A₁ + K₂A₂ + …)/d

Here, K represents the dielectric constant, A represents plate area and d represents plate separation.

A dielectric changes the electric field inside a capacitor. For a dielectric completely filling an isolated capacitor after the battery has been disconnected:

E = E₀/K

The electric field's change depends on the circuit conditions, including whether the battery remains connected.

Memory tip: Dielectric layers along the field direction are treated as series combinations; side-by-side regions are treated as parallel combinations.

Electrostatic Potential and Capacitance becomes easier to revise when you understand the relationship between electric field, potential, and potential energy and practise applying the relevant formulas. Pay attention to sign conventions, capacitor combinations, dielectric effects, and energy calculations while solving numerical questions. Physics Wallah’s Electrostatic Potential and Capacitance notes can help you revise these concepts and practise applying them to Physics problems.       

NEET Resources You Might Like: