Chapter 2 - Electrostatic Potential and Capacitance

Definition: Electrostatic potential at a point is defined as the work done per unit positive test charge in bringing it from infinity to that point against electrostatic forces, without acceleration. V = W / q0. If W is work done by external agent without kinetic energy change, then V is positive if work is done against field. SI unit is Volt: 1 Volt = 1 Joule / Coulomb.

Potential difference between two points A and B is work done per unit charge in moving test charge from A to B: Vb - Va = Wab / q0 = -∫(A to B) E·dl. The line integral of electric field. If you move charge along field direction, potential decreases. If you move against field, potential increases. Electric field always points from higher potential to lower potential, like water flows from higher to lower height.

Relation between field and potential: E = - dV/dr. Field is negative gradient of potential. If potential changes rapidly with distance, field is strong. If potential is constant (equipotential region), field zero.

Potential due to a point charge q at distance r: V = (1/4πϵ0) q / r = k q / r. Note it falls as 1/r, while field falls as 1/r². Potential is scalar, so sign matters: positive charge gives positive potential everywhere, negative charge gives negative potential. At infinity, we take potential zero as reference. Potential can be positive, negative or zero, but field magnitude always positive.

Example: Potential due to 5 μC at 0.3 m: V = 9e9 *5e-6 /0.3 = 150,000 V = 150 kV.

Potential is scalar, so potential due to system of charges is algebraic sum of potentials due to each charge individually: V = Σ k qi / ri. No vector addition needed, simple addition with sign. This makes potential much easier to calculate than field which needs vector sum.

Potential due to electric dipole: Dipole has +q and -q separated by 2a. At a point r distance from center, angle θ between r and dipole axis (p direction). Potential V = (1/4πϵ0) p cosθ / r² for r>>a. So V depends on cosθ.

On axial line θ=0° cosθ=1 V_axial = kp / r² maximum positive on side of +q, negative on side of -q. On equatorial line θ=90° cosθ=0 V_equatorial=0. So equatorial plane is zero potential plane for ideal dipole. This is very important result.

Potential energy of system of charges is work done in assembling charges from infinity to their positions. For two charges q1 q2 separated r, U = k q1 q2 / r. For three charges, U = k(q1q2/r12 + q1q3/r13 + q2q3/r23) sum over pairs. If charges like, U positive means work must be done against repulsion to assemble, energy stored positive. If unlike, U negative, field does work.

Potential energy of dipole in external uniform field E: U = -p·E = -pE cosθ. Minimum at θ=0° when p aligned with E stable equilibrium, maximum at θ=180° opposite unstable. Work done in rotating dipole from θ1 to θ2 is W = pE(cosθ1 - cosθ2).

Equipotential surface is surface where potential V is same everywhere. Properties:

1. No work is done in moving a charge along equipotential surface because ΔV=0 so W=qΔV=0. The charge can move without energy change, like moving horizontally on flat ground no work against gravity.
2. Electric field is always perpendicular to equipotential surface. If E had component along surface, it would do work moving charge along surface which contradicts no work property. So E ⊥ equipotential.
3. No two equipotential surfaces can intersect. If they intersect, at intersection point there would be two values of potential which is impossible.
4. Equipotential surfaces are closer in regions of strong field and farther in weak field. Because E = -dV/dr, for same ΔV if dr small, E large.
5. Shape: For point charge spherical shells centered at charge, for uniform field parallel planes perpendicular to field lines, for dipole not spherical.

Field lines and equipotentials are mutually perpendicular, forming a grid. Field lines show direction of E, equipotentials show constant V.

Conductors have free electrons, so they have special electrostatic properties:

1. Inside a conductor electrostatic field is zero. Why? If there were field inside, free electrons would move under force and current would flow until field neutralized. In electrostatic equilibrium no current, so inside E=0.
2. Potential is same everywhere inside and on surface of conductor. Conductor is equipotential because E=0 inside so no potential difference.
3. Electric field just outside charged conductor is perpendicular to surface and magnitude σ/ε0 where σ is surface charge density. Just outside E = σ/ε0 n_hat, inside zero. For plane sheet E=σ/2ε0 both sides, but conductor has charge only on one side so E=σ/ε0 one side.
4. Excess charge resides only on surface of conductor. No excess inside, because if excess inside, field would not be zero. All charge goes to outer surface.
5. Electrostatic shielding: Cavity inside conductor. If cavity has no charge, field inside cavity zero and potential constant. This is shielding. Faraday cage protects sensitive instruments from external fields. Car is Faraday cage, lightning strikes outside car safe inside.
6. If cavity contains charge +q, inner surface of cavity gets -q induced and outer surface gets +q. Field inside cavity due to charge inside cavity, field outside due to outer charge.
7. At sharp points of irregular conductor, surface charge density high and field strong. Corona discharge happens at sharp points. Lightning rods use this principle.

Dielectric is insulator placed between plates of capacitor to increase capacitance. Two types of molecules:

Polar molecules have permanent dipole moment even without external field. Example water H2O, HCl. Positive and negative centers of charge do not coincide. In absence of field dipoles randomly oriented due to thermal agitation net dipole moment zero.

Non-polar molecules have no permanent dipole moment without field. Positive and negative centers coincide. Example H2, O2, CO2 symmetric. In external field, positive center shifts along field and negative opposite, induced dipole forms.

When dielectric placed in external field E0, polarization happens: Polar molecules align partially along field, non-polar molecules get induced dipole moment. This creates bound charges on surfaces of dielectric slab - positive bound charge on side towards negative plate and negative bound charge on side towards positive plate. These bound charges create internal field Ep opposite to E0, so net field inside dielectric reduced: E = E0 / K where K = ε/ε0 is dielectric constant >1.

Dielectric constant K (also called relative permittivity εr) is ratio of capacitance with dielectric to without. K>1, for vacuum 1, for water 80, for glass 5-10, for mica 6. Polarization P is defined as dipole moment per unit volume. Relation: E = E0 - Ep.

Effect on capacitor: Field reduced so potential difference V = E d reduced for same charge Q, so capacitance C = Q/V increases K times: C = K C0. So dielectric increases capacitance.

Capacitor is device consisting of two conductors separated by insulator or dielectric that stores charge Q and energy. When battery connected, one conductor gets +Q other gets -Q. Potential difference V between conductors proportional to Q: Q = C V. Capacitance C = Q / V defined as charge per unit potential. SI unit Farad F: 1 F = 1 C / V. Farad is large, usually microfarad μF 10⁻⁶ F or picofarad pF 10⁻¹² F.

Parallel plate capacitor: Two parallel plates area A separation d vacuum. Field between plates E = σ/ε0 = Q/(A ε0). Potential difference V = E d = Q d / (A ε0). So C = Q/V = ε0 A / d for vacuum. With dielectric K completely filling, C = K ε0 A / d = K C0. Capacitance depends on geometry A d and dielectric, not on Q or V.

Combination of capacitors: Series: Capacitors connected end to end, charge on each same Q, total voltage V = V1+V2+... So 1/Ceq = 1/C1+1/C2+... Equivalent capacitance smaller than smallest individual. Parallel: Capacitors connected side by side, voltage same V on each, total charge Q = Q1+Q2+... So Ceq = C1+C2+... Equivalent larger than largest. Series used to reduce capacitance or increase voltage handling, parallel to increase capacitance.

When capacitor charged, work is done against electric field to move charge Q from one plate to other. This work is stored as electrical potential energy. Derive: Small charge dQ moved at potential V = Q/C, work dW = V dQ = Q/C dQ. Integrate 0 to Q: W = ∫ Q/C dQ = Q²/(2C) = ½ QV = ½ C V² = Q²/(2C). So energy stored U = ½ QV = ½ C V² = Q²/(2C) Joules.

Where is energy stored? In electric field between plates. Energy density u = energy per unit volume = ½ ε0 E² for vacuum, with dielectric ½ K ε0 E². So field contains energy.

Sharing of charges: When two charged capacitors C1 V1 Q1=C1V1 and C2 V2 Q2=C2V2 connected together positive to positive negative to negative, charge flows until common potential V_common reached. Total charge conserved Q_total=Q1+Q2=C1V1+C2V2. Common potential V_common = Q_total/(C1+C2) = (C1V1+C2V2)/(C1+C2). If V1≠V2 some energy lost as heat and spark: Energy loss = ½ C1C2 (V1-V2)²/(C1+C2). This loss is not violation of conservation, it goes as heat and electromagnetic radiation during charge flow.

Battery charging capacitor: Battery provides charge, builds Q on plates, stores energy U, but battery does work QV, half stored half lost as heat in resistance if charging through resistor.

Van de Graaff generator is device to produce high voltage of order million volts used to accelerate charged particles for experiments. Principle: Action of sharp points (corona discharge) and property that charge resides only on outer surface of conductor.

Construction: Large hollow metal sphere on insulating column, rubber belt moving over two rollers one at ground one inside sphere driven by motor, belt carries charge from ground to sphere inside. Sharp points near lower roller spray charge onto belt by corona discharge, belt carries charge up, another sharp points inside sphere collect charge and transfer to sphere. Charge accumulates on outer surface of sphere potential rises very high. Potential can be V = kQ/R, with large R and Q potential million volts.

Uses: To accelerate electrons, protons, ions to high energy for nuclear physics experiments, cancer treatment. Limitation: Sparking at high voltage, air breakdown ~3×10⁶ V/m.

We have now completed electrostatics foundation - potential and capacitance. Next chapter Current Electricity deals with moving charges.