Derive expressions for the electric field of an electric dipole at a point on (a) its axial line and (b) its equatorial line. Compare the two for a short dipole.

1. 1 · Define the dipole moment. p = q × 2a, a vector from −q to +q, unit C m. Let k = 1/4πε₀.
2. 2 · Draw both figures. [−q at A, +q at B, centre O; axial point P beyond B; equatorial point P on the bisector; fields E₊, E₋ and the resultant marked]
3. 3 · Axial line: write the two fields. E₊ = kq/(r − a)² along p; E₋ = kq/(r + a)² against p.
4. 4 · Axial line: add. E = kq × 4ar/(r² − a²)² = 2kpr/(r² − a²)²; for r ≫ a, E = 2kp/r³ along p.
5. 5 · Equatorial line: write and resolve. E₊ = E₋ = kq/(r² + a²). Perpendicular parts cancel; parallel parts add.
6. 6 · Equatorial line: result. E = 2kqa/(r² + a²)^(3/2) = kp/(r² + a²)^(3/2), opposite to p; for r ≫ a, E = kp/r³.
7. 7 · Compare. At the same large r, E(axial) : E(equatorial) = 2 : 1, and they point in opposite directions.
8. 8 · Close in one line. A dipole's field falls as 1/r³ and is twice as strong on its axis as on its equator.

Scoring tip: Write k = 1/4πε₀ once at the start. Then box both short-dipole results and state the 2 : 1 ratio.