Electric fields

Electric fields

1. What an Electric Field Is

Electric field $E$ at point $P$ = force per unit positive test charge placed at $P$ due to source charges, vector:

SI unit V/m (or N/C). It exists even without test charge; $F = q E$ for charge $q$ in field.

Visualized as field lines (tangent gives $E$ direction, density ∝ $|E|$).

2. Coulomb's Law — Point Charge

For point charge $Q$ at origin:

with $k=9×10^9$ N·m²/C², $ε_0=8.85×10^{-12}$ F/m. For many charges, superposition:

For continuous distribution:

3. Key Geometries

• Infinite line charge $λ$ (C/m):

(radial, ∝1/r)

• Infinite sheet $σ$ (C/m²):

(uniform, independent of distance)

• Parallel plate capacitor $±σ$:

uniform between plates.

4. Relation to Potential

$E$ is conservative for electrostatics:

where $V(r)= (1/4π ε_0) Q/r$ for point charge. Equipotentials ⊥ field lines.

5. Worked Examples

(a) E at 1 m from 1 nC: E=9e9·1e-9/1=9 V/m radially outward ✓

(b) Two +Q at (±a,0): E at origin cancels to 0 by symmetry; at (0,y): E_y =2·(kQ/(a²+y²))· y/√(a²+y²) ✓

(c) Field between plates V=10 V, d=1 mm: E=10/1e-3=10⁴ V/m ✓

(d) Uniformly charged ring axis: E= kQ z/(z²+R²)^{3/2} \hat{z}

6. Exam Points

• Define $E=F/q_0$ — 3 marks, frequent.
• Coulomb's law and superposition — 5 marks, very frequent.
• Sketch field lines for point, dipole, line — 3-5 marks, frequent.
• $E=-∇V$ — 3 marks.
• Compute $E$ for line/sheet/ring — 5-7 marks, frequent.