The distance between charges +q and -q is 2 l and between +2 q and -2 q is 4 l. The electrostatic potential at point P at a distance r from center O is -α[(q l)/r²] × 10⁹ V, where the value of α is __________. (Use 1/(4 π ε₀)=9 × 10⁹ Nm² C⁻²)

Solution

Physics sir · 4 steps

Treat the four charges as two opposite dipoles at O, add their moments, then use the dipole formula.

1. Two dipoles. The inner pair is -q at x=-l and +q at x=+l. A dipole moment points from the negative charge to the positive charge, so p₁ = q(2l) = 2ql points along +x. The outer pair is +2q at x=-2l and -2q at x=+2l, so p₂ = 2q(4l) = 8ql points along -x.
2. Net dipole moment. Both dipoles share the centre O and lie on the same line, so their moments add as vectors: p = 8ql - 2ql = 6ql, pointing along -x. P is far away (r ≫ l), so the whole arrangement acts as one dipole.

   V = 1/4πε₀ (p cos θ)/r²

3. Find the angle. θ is measured from the direction of p⃗, which is -x. OP makes 60° with +x, so θ = 180° - 60° = 120° and cos θ = -1/2.
4. So. V = 9 × 10⁹ × (6ql × (-1/2))/r² = -27[ql/r²] × 10⁹ V, so α = 27.

The common slip: 45. Adding the two moments as 2ql + 8ql = 10ql while ignoring that they point in opposite directions gives 45, not 27.

The idea to keep. Several dipoles at one centre act like a single dipole far away. Add their moments as vectors, taking each from negative to positive charge, and measure θ from that net direction.

Answer from the NTA final answer key. Solution by PrepWiser.