Two shorts dipoles (A,B),A having charges ± 2μC and length 1 cm and B having charges ± 4μC and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point P, equi-distant from the centres of both dipoles is ____ N/C.

Solution

Physics sir · 4 steps

P sits on the axis of A but on the equator of B, so the two fields come out equal and at right angles.

1. Where P sits. P is midway, so it is r = 40 cm = 0.4 m from each centre. P lies on the line of dipole A (its axial line) and on the perpendicular bisector of dipole B (its equatorial line). Both dipoles are short (1 cm ≪ 40 cm), so the short-dipole formulas apply.
2. Dipole moments. p_A = 2 × 10⁻⁶ × 0.01 = 2 × 10⁻⁸ C m and p_B = 4 × 10⁻⁶ × 0.01 = 4 × 10⁻⁸ C m. Also r³ = (0.4)³ = 0.064 m³.

   E_A = 2kp_A/r³ = (2 × 9 × 10⁹ × 2 × 10⁻⁸)/0.064 = 5625 N/C, E_B = kp_B/r³ = (9 × 10⁹ × 4 × 10⁻⁸)/0.064 = 5625 N/C

3. Directions. On the axis, the field points along p⃗_A, which is horizontal (towards the right). On the equator, the field points opposite to p⃗_B, which is vertical (downwards). So E_A and E_B are perpendicular.
4. So. E = √(E_A² + E_B²) = 5625√2 N/C. Since 9/16 = 0.5625, this is 9/16√2 × 10⁴ N/C, option (b).

The idea to keep. For a short dipole, the axial field 2kp/r³ is twice the equatorial field kp/r³ at the same distance. Always check whether the point is on the axis or the equator before choosing the formula.

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