Electrostatic Potential and Capacitance: NEET notes

1. Electrostatic potential

NCERT §2.1 to §2.3

• The Coulomb force is conservative: the work it does between two points depends only on the end points, not on the path taken.
• Potential energy difference: the work an external agent does to move a charge from R to P slowly (no acceleration) against the field. U_P − U_R = W_RP, and the field's own work is −W_RP.
• Only differences of potential energy matter, so a constant can be added freely. The usual choice puts zero potential energy at infinity.
• Electrostatic potential V at a point: the work done by an external agent, per unit positive test charge, to bring it from infinity to that point. SI unit volt; 1 V = 1 J C⁻¹.
• The work to take a charge q from R to P is q(V_P − V_R). The test charge is taken very small so it does not disturb the charges producing the field.
• Point charge Q at the origin: V(r) = Q/(4πε₀r). V is positive for Q > 0 and negative for Q < 0, and it falls as 1/r while the field falls as 1/r².
• V depends only on distance from a point charge, so it is the same at every point of a sphere centred on the charge.
• Worked value: 4 × 10⁻⁷ C gives V = 9 × 10⁹ × 4 × 10⁻⁷ / 0.09 = 4 × 10⁴ V at 9 cm; bringing 2 × 10⁻⁹ C there from infinity needs 8 × 10⁻⁵ J, whatever the path.

2. Potential due to a dipole

NCERT §2.4

• A dipole is charges +q and −q a distance 2a apart; its dipole moment p = q × 2a points from −q to +q.
• Far from the dipole (r ≫ a), V = p cos θ/(4πε₀r²), where θ is the angle between p and the line to the point. In vector form V = p·r̂/(4πε₀r²).
• On the axis (θ = 0° or 180°) V = ±p/(4πε₀r²); on the equatorial plane (θ = 90°) V = 0 everywhere.
• Dipole potential falls as 1/r², faster than a single charge's 1/r, because the two opposite charges nearly cancel at a distance.
• The dipole potential depends on direction as well as distance: it is not spherically symmetric, unlike a point charge's.
• Zero potential on the equatorial plane does not mean zero field there: E on the equator is p/(4πε₀r³), pointing opposite to p.

3. Potential due to a system of charges

NCERT §2.5

• Superposition holds for potential: V at a point is the sum of the potentials of each charge on its own, V = (1/4πε₀) Σ qᵢ/rᵢ.
• Potential is a scalar, so the sum is plain algebra with signs; no directions or components are needed. This is why potential is often easier to find than field.
• A continuous distribution is split into small elements, each treated as a point charge, and the contributions are added (integrated).
• Uniformly charged spherical shell of charge q and radius R: outside, V = q/(4πε₀r), as if all the charge sat at the centre.
• Inside the shell E = 0, so no work is done moving a test charge inside, and V stays at its surface value q/(4πε₀R) throughout.
• Worked value: 3 × 10⁻⁸ C and −2 × 10⁻⁸ C placed 15 cm apart give V = 0 at two points on their line: 9 cm and 45 cm from the positive charge.
• Points of zero potential are not points of zero field: at 9 cm between the two charges above, both fields point the same way.

4. Equipotential surfaces

NCERT §2.6

• An equipotential surface is one on which V has the same value at every point. Moving a charge along it needs no work.
• The field is everywhere at right angles to the equipotential surface through that point; if it had a component along the surface, moving along it would take work.
• Shapes: concentric spheres around a point charge; planes at right angles to the field in a uniform field; distorted closed surfaces around a dipole or two like charges.
• Field and potential: E = −δV/δl, so the size of E is the drop in potential per unit distance taken at right angles to the equipotentials.
• E points in the direction in which V decreases fastest (the steepest fall of potential).
• Where equipotentials drawn at equal steps of V crowd together the field is strong; where they spread apart it is weak.
• In a uniform field E, two planes a distance d apart along the field differ in potential by Ed.

5. Potential energy of a system of charges

NCERT §2.7

• The potential energy of a system of charges is the total external work needed to bring them from infinity, one by one, to their places.
• Two charges: U = q₁q₂/(4πε₀r₁₂). The first charge costs nothing; the second costs q₂ times the potential of the first.
• Like charges (q₁q₂ > 0) give positive U, since work must be done against repulsion. Unlike charges give negative U, since work is needed to pull them apart.
• Three charges: U = (1/4πε₀)(q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃), one term for every pair.
• Because the force is conservative, U depends only on the final arrangement and not on the order or paths used to assemble it.
• Worked value: +q, −q, +q, −q at the corners of a square of side d (alternating) have U = −(q²/4πε₀d)(4 − √2). The centre of that square is at V = 0, so a further charge can be brought there for no work.

6. Potential energy in an external field

NCERT §2.8

• Here the field comes from outside sources, which the charge in question is assumed not to disturb. Only the charge's energy in that field is counted, not the energy of the sources.
• A single charge q at a point where the external potential is V(r) has potential energy qV(r).
• Electron volt: the energy gained by an electron moved through 1 V. 1 eV = 1.6 × 10⁻¹⁹ J; 1 keV = 1.6 × 10⁻¹⁶ J, 1 MeV = 1.6 × 10⁻¹³ J.
• Two charges in an external field: U = q₁V(r₁) + q₂V(r₂) + q₁q₂/(4πε₀r₁₂), the energy of each in the outside field plus their mutual energy.
• Worked value: 7 μC and −2 μC held 18 cm apart on the x-axis, 9 cm either side of the origin, have mutual energy −0.7 J, so separating them completely takes 0.7 J. Put them in an outside field E = A/r² (A = 9 × 10⁵ N C⁻¹ m², V = A/r): the charges gain 70 J and −20 J, and the total is 70 − 20 − 0.7 = 49.3 J.
• Dipole in a uniform field E: U(θ) = −pE cos θ = −p·E, taking U = 0 at θ = 90°. Turning it from θ₀ to θ₁ takes work pE(cos θ₀ − cos θ₁).
• U is least (−pE) when p is along E, the stable position, and greatest (+pE) when p is opposite to E, an unstable one.
• Worked value: a mole of dipoles, each 10⁻²⁹ C m, fully aligned in 10⁶ V m⁻¹ has U = −6 J. Turn the field by 60° and U rises to −3 J as the dipoles realign, so 3 J comes out as heat.

7. Electrostatics of conductors

NCERT §2.9

• Metals have free electrons that can move inside the metal but cannot leave it. In a field they drift opposite to the field until the field inside falls to zero.
• In the static state the electric field inside a conductor is zero everywhere, whether it is neutral or charged and whatever the outside field.
• Just outside a charged conductor the field is normal to the surface at each point; a component along the surface would push the surface charges.
• A conductor can hold no excess charge in its interior; Gauss's law on any small closed surface inside gives zero enclosed charge, so all excess charge sits on the surface.
• The whole conductor, inside and on its surface, is at one potential. Different conductors in a system can be at different potentials.
• Field just outside a charged conductor: E = (σ/ε₀) n̂, where σ is the local surface charge density and n̂ the outward normal (inward for σ < 0).
• Electrostatic shielding: a cavity inside a conductor, with no charge in it, has zero field whatever the charges and fields outside. This protects sensitive instruments.
• Shielding works only one way: a charge placed inside a cavity still produces a field outside the conductor.
• Everyday cases: aircraft tyres made slightly conducting and fuel tankers trailing metal chains let friction charge drain to earth before it can spark.

8. Dielectrics and polarisation

NCERT §2.10

• Dielectrics are insulators with no (or negligible) free charge. A field cannot drive charges through them, but it can shift or turn the charges inside molecules.
• A conductor cancels an applied field inside it completely; a dielectric only weakens it.
• Non-polar molecules (O₂, H₂) have their positive and negative charge centres together and no permanent dipole moment. A field pulls the centres apart and induces a dipole along the field.
• Polar molecules (H₂O, HCl) have a permanent dipole moment. Without a field they point every which way and cancel; a field partly lines them up, against the scrambling of thermal motion.
• Polarisation P is the dipole moment per unit volume. For a linear isotropic dielectric P = χₑε₀E, where χₑ is the electric susceptibility.
• Inside a uniformly polarised slab the dipole ends cancel, but the two faces normal to the field carry bound surface charges +σₚ and −σₚ.
• The field of these bound surface charges opposes the applied field, so the net field inside the dielectric is smaller.

9. Capacitors and the parallel plate capacitor

NCERT §2.11 and §2.12

• A capacitor is two conductors separated by an insulator, usually carrying charges +Q and −Q with a potential difference V between them. Q is the charge on one plate; the total is zero.
• Doubling Q doubles the field and so doubles V, so Q/V is fixed: C = Q/V, the capacitance. It depends on the shape, size and spacing of the conductors and on the insulator between them, not on Q or V.
• Unit: 1 farad = 1 C V⁻¹ = 1 C² N⁻¹ m⁻¹. The farad is huge; practical capacitors are in μF (10⁻⁶ F), nF (10⁻⁹ F) and pF (10⁻¹² F).
• A large C stores a large charge at a small V. This matters because a high V means a strong field that can ionise the air and let the charge leak away.
• Dielectric strength is the largest field an insulator withstands before it breaks down; for air it is about 3 × 10⁶ V m⁻¹, which is 3 × 10⁴ V across a 1 cm gap.
• Parallel plate capacitor (plate area A, gap d, d² ≪ A): outside the plates the fields of the two sheets cancel, and between them they add to E = σ/ε₀ = Q/(ε₀A), uniform and pointing from + to −.
• V = Ed = Qd/(ε₀A), so C = ε₀A/d. Near the plate edges the field lines bulge outward (fringing), which this formula ignores.
• Worked values: A = 1 m² and d = 1 mm give C = 8.85 × 10⁻⁹ F. For 1 F with d = 1 cm the plates would need an area of about 10⁹ m², roughly 30 km on each side.

10. Effect of dielectric on capacitance

NCERT §2.13

• With vacuum between the plates, E₀ = σ/ε₀, V₀ = E₀d and C₀ = ε₀A/d.
• A dielectric filling the gap is polarised; its bound surface charges ±σₚ cut the effective charge density to σ − σₚ, so E = (σ − σₚ)/ε₀.
• For a linear dielectric σ − σₚ = σ/K, where K > 1 is the dielectric constant. The field and V both fall by the factor K for the same charge.
• Hence C = Kε₀A/d = KC₀. The dielectric constant is the factor by which capacitance grows when the dielectric fills the space fully.
• Permittivity of the medium ε = ε₀K; K = ε/ε₀ is dimensionless. For vacuum K = 1.
• C = KC₀ holds for a capacitor of any shape, so it can serve as the definition of K.
• Slab partly filling the gap: a slab of constant K and thickness 3d/4 (full plate area) makes V = E₀d(K + 3)/(4K) for the same charge, so C = 4KC₀/(K + 3).

11. Combination of capacitors

NCERT §2.14

• Series: capacitors joined end to end. Every capacitor carries the same charge Q, because the joined inner plates are an isolated conductor with zero net charge.
• In series the voltages add: V = Q/C₁ + Q/C₂ + …, so 1/C = 1/C₁ + 1/C₂ + … + 1/Cₙ.
• The series combination is smaller than the smallest capacitor in it, and the smallest capacitor takes the largest share of the voltage.
• Parallel: all capacitors share the same potential difference V, and each takes its own charge Qᵢ = CᵢV.
• In parallel the charges add, Q = Q₁ + Q₂ + …, so C = C₁ + C₂ + … + Cₙ.
• Worked value: three 10 μF capacitors in series (10/3 μF) in parallel with a fourth 10 μF give 13.3 μF. On 500 V the series three each hold 1.7 × 10⁻³ C and the fourth holds 5.0 × 10⁻³ C.
• The capacitor rules are the reverse of the resistor rules: series capacitors combine like parallel resistors.

12. Energy stored in a capacitor

NCERT §2.15

• Charging moves charge from the negative plate to the positive plate, which is at the higher potential, so external work is done at every step.
• When the charge is Q′ the voltage is Q′/C; moving a further δQ′ takes (Q′/C)δQ′. Adding these from 0 to Q gives W = Q²/(2C).
• Stored energy U = Q²/(2C) = ½CV² = ½QV. It is ½QV, not QV, because the voltage builds up from zero while the charge is moved.
• The energy can be pictured as stored in the field between the plates. For a parallel plate capacitor U = ½ε₀E² × Ad, where Ad is the volume of the gap.
• Energy density of an electric field u = ½ε₀E² (joules per cubic metre). This holds for any field, not only a capacitor's.
• Worked value: 900 pF charged to 100 V holds Q = 9 × 10⁻⁸ C and U = 4.5 × 10⁻⁶ J.
• Join that charged capacitor to an identical uncharged one: the charge shares equally, V halves to 50 V, and the total energy drops to 2.25 × 10⁻⁶ J. Charge is conserved; the missing half is lost as heat and electromagnetic radiation while charge flows.

Must-know facts

1. V = Q/(4πε₀r) for a point charge: it falls as 1/r, the field as 1/r².
2. Potential is a scalar: add the potentials of several charges with their signs, no components.
3. Work to move q from A to B = q(V_B − V_A), independent of path.
4. Dipole: V = p cos θ/(4πε₀r²); zero everywhere on the equatorial plane; falls as 1/r².
5. Charged shell: V is constant inside and equal to its surface value q/(4πε₀R); E inside is zero.
6. E is perpendicular to equipotentials and points towards decreasing V; E = −dV/dl.
7. No work is done moving a charge along an equipotential surface.
8. Two charges: U = q₁q₂/(4πε₀r); positive for like charges, negative for unlike.
9. Dipole in a uniform field: U = −pE cos θ; minimum at θ = 0°, maximum at 180°.
10. 1 eV = 1.6 × 10⁻¹⁹ J.
11. Inside a conductor in the static state: E = 0, no excess charge, V the same everywhere.
12. Field just outside a conductor = σ/ε₀, normal to the surface.
13. C = Q/V; parallel plate C = ε₀A/d; with a dielectric filling the gap C = Kε₀A/d.
14. Dielectric strength of air ≈ 3 × 10⁶ V m⁻¹.
15. Series: 1/C = Σ1/Cᵢ, same Q on each. Parallel: C = ΣCᵢ, same V on each.
16. U = ½CV² = Q²/(2C) = ½QV; energy density ½ε₀E².
17. Sharing charge between two capacitors conserves charge but loses energy.
18. Non-polar: O₂, H₂. Polar: H₂O, HCl.

Common traps

Formulas

Key terms

Conservative force

A force whose work between two points does not depend on the path, so a potential energy can be defined for it.

Electrostatic potential

External work per unit positive test charge to bring it slowly from infinity to a point; measured in volts.

Potential difference

Work per unit positive charge to move a charge from one point to another; the physically meaningful quantity.

Volt

One joule of work per coulomb of charge.

Equipotential surface

A surface on which the potential has one value everywhere; the field crosses it at right angles.

Electrostatic potential energy

The external work needed to assemble a set of charges from infinity into their places.

Electron volt

Energy gained by an electron moved through a potential difference of 1 V; 1.6 × 10⁻¹⁹ J.

Electrostatic shielding

The zero field inside an empty cavity of a conductor, whatever charges or fields lie outside.

Dielectric

An insulator with no free charges, which a field can polarise but not drive a current through.

Polar molecule

A molecule whose positive and negative charge centres are apart, giving it a permanent dipole moment.

Non-polar molecule

A molecule whose charge centres coincide, so it has no dipole moment until a field induces one.

Polarisation

Dipole moment per unit volume of a dielectric in a field.

Electric susceptibility

The constant χₑ linking polarisation to field in a linear dielectric, P = χₑε₀E.

Capacitance

Charge stored per unit potential difference between the two conductors of a capacitor.

Dielectric strength

The largest field an insulator can bear before it breaks down and conducts.

Fringing

The outward bulge of field lines near the edges of a capacitor's plates.

Dielectric constant

The factor K by which a dielectric filling the gap multiplies a capacitor's capacitance; K = ε/ε₀.

Energy density

Energy stored per unit volume of an electric field, ½ε₀E².