CBSE Class 12 Physics Electrostatic Potential and Capacitance Notes Set 03

1. Electrostatic Potential and Potential Difference

The electrostatic force is a conservative force: the work it does on a charge moving between two points depends only on the end points, not on the path taken. So we can define a potential energy, and from it a potential.

(i) Potential difference: The potential difference between points A and B is the work done by an external agent in moving a unit positive charge from A to B slowly (without acceleration), against the electrostatic force.
V_B − V_A = \( \frac{W_{AB}}{q} \)
(ii) Electrostatic potential at a point: the work done by an external agent in bringing a unit positive test charge from infinity to that point (taking V = 0 at infinity).
V = \( \frac{W_{\infty \to P}}{q} \)
(iii) Potential is a scalar. SI unit: volt (V), where 1 V = 1 J C⁻¹. Dimensions: [M L² T⁻³ A⁻¹].
(iv) Only differences of potential are physically meaningful; the zero of potential can be chosen anywhere (usually at infinity).

2. Potential due to a Point Charge

The potential at distance r from a point charge q is
V = \( \frac{1}{4\pi\varepsilon_0} \frac{q}{r} \)
1. V is positive near a positive charge and negative near a negative charge.
2. V varies as 1/r, while the field E varies as 1/r².
3. A positive test charge moves naturally from higher to lower potential; an electron moves from lower to higher potential.

Charged spherical shell or conducting sphere (charge q, radius R): outside (r ≥ R), V = \( \frac{1}{4\pi\varepsilon_0} \frac{q}{r} \); inside (r < R), V = \( \frac{1}{4\pi\varepsilon_0} \frac{q}{R} \), the same constant value as on the surface, because E = 0 inside.

3. Potential due to an Electric Dipole

For a short dipole of moment p, at a point at distance r from its centre (r ≫ a), where the line to the point makes angle θ with p:
V = \( \frac{1}{4\pi\varepsilon_0} \frac{p \cos\theta}{r^2} \) = \( \frac{1}{4\pi\varepsilon_0} \frac{\vec{p} \cdot \hat{r}}{r^2} \)
1. On the axial line (θ = 0° or 180°): V = ± \( \frac{1}{4\pi\varepsilon_0} \frac{p}{r^2} \)
2. On the equatorial line (θ = 90°): V = 0, although the field there is not zero.
3. The dipole potential falls as 1/r², faster than the 1/r potential of a single charge. It depends on the angle θ as well as on r.

4. Potential due to a System of Charges

By the superposition principle, the potential at a point due to charges q₁, q₂, …, q_n at distances r₁, r₂, …, r_n is the algebraic sum of their individual potentials:
V = \( \frac{1}{4\pi\varepsilon_0} \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} + \ldots + \frac{q_n}{r_n} \right) \)
Since potential is a scalar, no directions are involved; only the signs of the charges matter. This makes potential easier to calculate than field.

5. Equipotential Surfaces

A surface on which the potential has the same value at every point is called an equipotential surface.

Properties of equipotential surfaces:
1. No work is done in moving a charge between any two points on an equipotential surface (W = qΔV = 0).
2. The electric field is always perpendicular to the equipotential surface at every point.
3. The field points from higher to lower potential.
4. Two equipotential surfaces never intersect, since a point cannot have two potentials.
5. Where equipotentials (drawn for equal steps of V) are closer together, the field is stronger.
Shapes: concentric spheres for a point charge; planes perpendicular to the field for a uniform field; the surface of any conductor in electrostatic equilibrium is an equipotential.

6. Relation between Electric Field and Potential

The field is the negative of the potential gradient:
E = \( -\frac{dV}{dr} \) (along the direction of the field), and in a uniform field E = \( \frac{V}{d} \) in magnitude.
1. The electric field points in the direction in which the potential decreases most steeply.
2. Its magnitude equals the change of potential per unit displacement normal to the equipotential surface.
3. E = 0 in a region does not mean V = 0 there; it means V is constant (for example, inside a charged conductor).
4. V = 0 at a point does not mean E = 0 there (for example, the midpoint between +q and −q).

7. Potential Energy of a System of Charges

The potential energy of a system of point charges is the work done in assembling them by bringing them from infinity to their positions.
(i) Two charges q₁ and q₂ separated by r₁₂:
U = \( \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r_{12}} \)
U is positive for like charges (work must be done to push them together) and negative for unlike charges (they attract).
(ii) Three charges: U = \( \frac{1}{4\pi\varepsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_2 q_3}{r_{23}} + \frac{q_1 q_3}{r_{13}} \right) \), one term for every pair.
The energy does not depend on the order in which the charges are assembled.

8. Potential Energy in an External Field

(i) Single charge: A charge q at a point where the external potential is V has potential energy U = qV.
An electron accelerated through a potential difference of 1 V gains energy 1 eV = 1.6 × 10⁻¹⁹ J.
(ii) Two charges in an external field: U = q₁V(r₁) + q₂V(r₂) + \( \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r_{12}} \)
(iii) Dipole in a uniform field E at angle θ to the field:
U(θ) = −pE cos θ = \( -\vec{p} \cdot \vec{E} \)
1. θ = 0°: U = −pE (minimum, stable equilibrium).
2. θ = 90°: U = 0.
3. θ = 180°: U = +pE (maximum, unstable equilibrium).
Work done in rotating the dipole from θ₁ to θ₂: W = pE (cos θ₁ − cos θ₂). For example, turning it from 0° to 180° needs W = 2pE.

9. Electrostatics of Conductors

A conductor contains free charges (electrons in metals) that can move through the material. In an insulator (dielectric) the charges are bound to their atoms or molecules and can only be displaced slightly. In electrostatic equilibrium a conductor has these properties:
1. The electrostatic field inside the conductor is zero.
2. Just outside the surface, the field is normal to the surface at every point.
3. There is no net (excess) charge in the interior; any excess charge resides on the surface.
4. The potential is constant throughout the volume of the conductor and equal to its value on the surface.
5. The field just outside the surface is E = \( \frac{\sigma}{\varepsilon_0} \) \( \hat{n} \), where σ is the local surface charge density.
6. Electrostatic shielding: The field inside a cavity of a conductor (with no charge in the cavity) is zero, whatever the charges and fields outside. Sensitive instruments are protected by enclosing them in metal cases, and a car is a safe place during a thunderstorm for this reason.

10. Dielectrics and Polarisation

A dielectric is an insulating substance. When placed in an external field E₀, its molecules develop or align dipole moments; this is called polarisation.
(i) Non-polar molecules (O₂, N₂, H₂): the centres of positive and negative charge coincide; the field displaces them slightly and induces a dipole moment.
(ii) Polar molecules (H₂O, HCl): they have a permanent dipole moment; the field tends to align the randomly oriented dipoles.
(iii) Polarisation P is the dipole moment per unit volume. For a linear isotropic dielectric, P = χ_eE, where χ_e is the electric susceptibility.
(iv) The polarised dielectric develops bound surface charges that set up a field opposing E₀. The net field inside is reduced: E = \( \frac{E_0}{K} \).
(v) Dielectric constant K = \( \frac{\varepsilon}{\varepsilon_0} \) = 1 + χ_e, a pure number. K = 1 for vacuum, about 80 for water, and effectively infinite for a metal.

11. Capacitors and Capacitance

A capacitor is a system of two conductors separated by an insulator, used to store charge and electrical energy. The conductors carry charges +Q and −Q, and there is a potential difference V between them.
Q ∝ V, so Q = CV, and C = \( \frac{Q}{V} \)
1. The capacitance C depends only on the shape, size and separation of the conductors and on the medium between them, not on Q or V.
2. SI unit: farad (F), 1 F = 1 C V⁻¹. Practical units: μF (10⁻⁶ F), nF (10⁻⁹ F), pF (10⁻¹² F).
3. The maximum field a dielectric can withstand without breaking down is its dielectric strength (about 3 × 10⁶ V m⁻¹ for air). This limits the voltage a capacitor can safely hold.

12. Parallel Plate Capacitor

Two large plane parallel conducting plates, each of area A, separated by a small distance d. With charges ±Q, the surface charge density is σ = Q/A.
1. Between the plates the field is uniform: E = \( \frac{\sigma}{\varepsilon_0} \) = \( \frac{Q}{\varepsilon_0 A} \). Outside the plates E = 0 (fringing near the edges is neglected).
2. Potential difference: V = Ed = \( \frac{Qd}{\varepsilon_0 A} \)
3. Capacitance (vacuum or air between the plates):
C₀ = \( \frac{\varepsilon_0 A}{d} \)
C increases with plate area and decreases with separation.

13. Effect of a Dielectric on Capacitance

(i) If the space between the plates is completely filled with a dielectric of dielectric constant K:
C = \( \frac{K \varepsilon_0 A}{d} \) = K C₀
The field and the potential difference fall to 1/K of their values for the same charge, so the capacitance rises K times.
(ii) A dielectric slab of thickness t (t < d) and the same area as the plates:
C = \( \frac{\varepsilon_0 A}{d - t + \frac{t}{K}} \). A metal slab (K → ∞) of thickness t gives C = \( \frac{\varepsilon_0 A}{d - t} \).
(iii) Inserting a dielectric (K) completely:

14. Combination of Capacitors

(i) Series combination: Each capacitor carries the same charge Q; the total potential difference is shared, V = V₁ + V₂ + V₃.
\( \frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \)
For two capacitors, C_s = \( \frac{C_1 C_2}{C_1 + C_2} \). The equivalent capacitance is less than the smallest individual capacitance. In series, V ∝ 1/C: the smallest capacitor takes the largest share of the voltage.
(ii) Parallel combination: Each capacitor has the same potential difference V; the total charge is Q = Q₁ + Q₂ + Q₃.
C_p = C₁ + C₂ + C₃
The equivalent capacitance is greater than the largest individual capacitance. In parallel, Q ∝ C.
(iii) For n identical capacitors C: C_s = C/n and C_p = nC, so C_p/C_s = n².

15. Energy Stored in a Capacitor

Work done in charging a capacitor is stored as electrostatic potential energy in the field between its plates:
U = \( \frac{1}{2} CV^2 \) = \( \frac{Q^2}{2C} \) = \( \frac{1}{2} QV \)
1. Energy density (energy per unit volume of the field region): u = \( \frac{1}{2} \varepsilon_0 E^2 \). This holds for any electric field, not only that of a capacitor.
2. For capacitors in series (same Q), U = Q²/2C, so the smaller capacitor stores more energy. In parallel (same V), U = ½CV², so the larger capacitor stores more energy.
3. Sharing of charge: When two charged capacitors are connected (positive to positive), charge flows until both reach the common potential V = \( \frac{C_1 V_1 + C_2 V_2}{C_1 + C_2} \). Some energy is always lost as heat in the connecting wires: ΔU = \( \frac{C_1 C_2 (V_1 - V_2)^2}{2(C_1 + C_2)} \).

16. Quick Formula Recap