Lesson 13 of 13 · 17 min

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Must-know facts

18 facts

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

Common traps

Where marks are lost

Adding the potentials of several charges as vectors, or ignoring their signs.

Potential is a scalar. Write each qᵢ/rᵢ with the sign of its charge and add them as plain numbers.

Assuming the field is zero wherever the potential is zero (or the reverse).

E depends on how V changes, not on its value. On a dipole's equatorial plane V = 0 but E ≠ 0; inside a charged shell E = 0 but V ≠ 0.

Using 1/r for a dipole's potential.

A dipole's potential falls as 1/r², one power faster than a single charge. Its field falls as 1/r³.

Giving unlike charges a positive potential energy.

U = q₁q₂/(4πε₀r) carries the signs. Unlike charges give U < 0: work must be supplied to separate them.

Taking the stable position of a dipole as U = 0.

With U = −pE cos θ, the zero is at θ = 90°. The stable position θ = 0° has the lowest energy, −pE.

Inserting a dielectric with the battery removed and saying the charge rises by K.

Battery removed: Q is fixed, so V and E fall by K, C rises by K and U = Q²/2C falls by K. Battery connected: V is fixed, so Q, C and U all rise by K.

Using the parallel-resistor rule for parallel capacitors.

Capacitors add directly in parallel (C = C₁ + C₂) and as reciprocals in series, the opposite of resistors.

Writing the stored energy as QV.

The voltage rises from 0 to V during charging, so the average is V/2 and U = ½QV.

Thinking electrostatic shielding also keeps an inside charge's field from reaching outside.

A conductor shields its empty cavity from outside fields. A charge placed inside the cavity still produces a field outside.

Formulas

16 to know

Potential difference and work

W = q(V_P − V_R)

Work by an external agent to move q slowly from R to P; independent of path.

Potential of a point charge

V = Q/(4πε₀r)

Zero at infinity; 1/(4πε₀) = 9 × 10⁹ N m² C⁻².

Potential of a dipole

V = p cos θ/(4πε₀r²)

For r ≫ a; θ measured from p.

Superposition of potential

V = (1/4πε₀) Σ qᵢ/rᵢ

Scalar sum with signs.

Field from potential

E = −δV/δl

δl measured at right angles to the equipotential; E points towards falling V.

Energy of two charges

U = q₁q₂/(4πε₀r₁₂)

Add one such term for every pair in a larger system.

Energy of a charge in an external field

U = qV(r)

V is the potential of the external sources only.

Dipole in a uniform field

U = −p·E = −pE cos θ

Zero at θ = 90°; work to turn from θ₀ to θ₁ is pE(cos θ₀ − cos θ₁).

Field at a conductor's surface

E = σ/ε₀

Normal to the surface, outward for σ > 0.

Polarisation

P = χₑε₀E

Linear isotropic dielectric; χₑ is the susceptibility.

Capacitance

C = Q/V

1 F = 1 C V⁻¹.

Parallel plate capacitor

C = ε₀A/d

ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻²; with a dielectric filling the gap, C = Kε₀A/d.

Dielectric constant

K = C/C₀ = ε/ε₀

Dimensionless, greater than 1.

Series combination

1/C = 1/C₁ + 1/C₂ + … + 1/Cₙ

Same charge on each.

Parallel combination

C = C₁ + C₂ + … + Cₙ

Same voltage across each.

Energy stored

U = ½CV² = Q²/(2C) = ½QV

Energy density of a field u = ½ε₀E².

Key terms

18 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.

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