A point charge +Q is placed at a distance r from the center of a short electric dipole of dipole moment p on its axis. The force experienced by the charge is F₁ . The short dipole is then replaced by an extended dipole consisting of charges +q and -q separated by a distance 2a , having the same dipole moment p . The force on +Q at the same position is now F₂ . If the ratio F₂ F₁ = 16 9 , the value of the ratio r a is

Step-by-step solution

The force on a point charge +Q due to a short dipole on its axis is: F₁ = Q ( 1 4 ₀ 2p r^3 ) The force on +Q due to an extended dipole on its axis is given by the exact formula: F₂ = Q ( 1 4 ₀ 2pr (r^2-a^2)^2 ) Given the ratio F₂ F₁ = 16 9 : 2pr (r^2-a^2)^2 2p r^3 = 16 9 r^4 (r^2-a^2)^2 = 16 9 Taking the square root of both sides (since r > a ): r^2 r^2-a^2 = 4 3 Cross-multiplying yields: 3r^2 = 4r^2 - 4a^2 r^2 = 4a^2 Taking the square root again: r = 2a r a = 2 Answer: 2