Short on time? This page is the recap. For the derivations, Figures 1 and 2 and the worked examples, use the full study guide. This topic is in the calculus-based course only.
Recap
- A continuous charge is a sum of tiny point charges dq. Each gives dE = k dq/r², and E = ∫ k dq/r² r̂ (superposition).
- E is a vector. Integrate components, not magnitudes, unless every dE points the same way.
- For a line or arc, dq = λ dx or dq = λR dθ. Uniform rod: λ = Q/L. Uniform semicircle: λ = Q/(πR).
- Write r and every cos θ or sin θ in the one variable you integrate over.
- Symmetry tells you the direction and which components cancel.
- k = 1/(4πε₀) = 8.99 × 10⁹ N·m²/C²; ε₀ = 8.85 × 10⁻¹² C²/(N·m²).
Key relationships
| Distribution | Point | Field | Direction (positive charge) |
|---|---|---|---|
| Finite rod, Q, length L | On its line, d from near end | kQ/[d(d + L)] | Along the line, away from the rod |
| Finite rod, Q, length L | Perpendicular bisector, distance x | kQ/[x√(x² + L²/4)] | Perpendicular to the rod |
| Infinite line or long cylinder, λ | Distance r from axis | λ/(2πε₀r) = 2kλ/r | Radially outward |
| Ring, Q, radius a | Axis, distance z | kQz/(z² + a²)^(3/2); zero at centre; maximum at z = a/√2 | Along the axis |
| Semicircle, λ, radius R | Centre | 2kλ/R = 2kQ/(πR²) | Along the symmetry line, away from the arc |
| Quarter circle, λ, radius R | Centre | √2 kλ/R | Along the arc’s symmetry line (45° to each end radius), away from the arc |
Assumptions behind the results
- The objects are thin: all charge lies on a line or arc.
- The charge is uniform unless a density function is given.
- “Infinite” means the object is much longer than its distance from P.
Mistakes to avoid
- Adding magnitudes of dE vectors that point in different directions.
- Point-charge shortcut close up. kQ/r² from the centre is only valid far away.
- Two variables in one integral. Express r and the angle in x (or θ).
- λ = Q/(2πR) for a semicircle. Its length is πR.
- 1/r² for a long line. It is 1/r.
- Skipping the check. Test the far-away limit and the direction.
Quick self-check
- What is E at the centre of a uniformly charged full ring? (Zero: every piece is cancelled by the one opposite it)
- A semicircle of radius 0.20 m carries λ = 2.0 × 10⁻⁸ C/m. Find E at its centre. (2kλ/R = 1.8 × 10³ N/C)
- A finite rod’s field on its bisector is kQ/[x√(x² + L²/4)]. What does it become when L is much larger than x? (2kλ/x, the infinite-line field)
Next: practice questions.