Electric field and dipoles

Electric field, field lines and dipoles

The electric field at a point is the force per unit positive test charge placed there: E = F/q₀. Its SI unit is N/C (equal to V/m). For a point charge q, E = kq/r², pointing away from a positive charge and towards a negative charge.

Field lines are drawn so that the tangent at any point gives the direction of E. They start on positive charges and end on negative charges, never cross each other, and are crowded where the field is strong. Electrostatic field lines do not form closed loops.

An electric dipole is a pair of equal and opposite charges, +q and –q, separated by a small distance 2a. Its dipole moment is p = q × 2a, a vector pointing from –q to +q; unit C m.

For a short dipole at a large distance r:

• On the axial line: E = 2kp/r³ (along p).
• On the equatorial line: E = kp/r³ (opposite to p).

So the dipole field falls off as 1/r³, faster than the 1/r² of a single charge.

In a uniform field E, a dipole feels zero net force but a torque τ = pE sin θ, where θ is the angle between p and E. The torque tries to line p up with E.

Worked example: a dipole of moment 2 × 10⁻⁸ C m makes 30° with a field of 5 × 10⁴ N/C. τ = 2 × 10⁻⁸ × 5 × 10⁴ × 0.5 = 5 × 10⁻⁴ N m.

Key points

• E = kq/r² for a point charge.
• p = q(2a), directed from –q to +q.
• Axial field = 2 × equatorial field at the same distance.
• τ = pE sin θ; maximum at θ = 90°, zero at 0° and 180°.
• Net force on a dipole in a uniform field is zero.

Exam tips

• Expect quick substitution into E = kq/r² and τ = pE sin θ.
• Learn the 1/r³ dependence; ratio questions test it often.

Common mistakes

• Taking the dipole direction from + to –; it is from – to +.
• Thinking a dipole in a uniform field moves along the field; it only rotates.
• Believing two field lines can cross.