Lesson 13 of 13 · 17 min
Chapter review
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Animated walk through potential and capacitors
Must-know facts
18 facts
- 01V = Q/(4πε₀r) for a point charge: it falls as 1/r, the field as 1/r².
- 02Potential is a scalar: add the potentials of several charges with their signs, no components.
- 03Work to move q from A to B = q(V_B − V_A), independent of path.
- 04Dipole: V = p cos θ/(4πε₀r²); zero everywhere on the equatorial plane; falls as 1/r².
- 05Charged shell: V is constant inside and equal to its surface value q/(4πε₀R); E inside is zero.
- 06E is perpendicular to equipotentials and points towards decreasing V; E = −dV/dl.
- 07No work is done moving a charge along an equipotential surface.
- 08Two charges: U = q₁q₂/(4πε₀r); positive for like charges, negative for unlike.
- 09Dipole 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
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.