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A short electric dipole with dipole moment P (directed along positive y-axis) is placed at point (0, b) in the xy-plane. A point charge Q is placed at the origin O. The electric field at point A(a, a + b) is zero. The ratio b/a equals:

1. 2^(1/3)
2. 2^(2/3)
3. 2^(1/3) / 2
4. 1 / 2^(1/3)

Correct answer: 2^(1/3)

Solution

The dipole at (0,b) has moment P along +y. Point A is at (a, a+b). Relative to the dipole, A is at position vector r_vec = (a, a), so r = a*sqrt(2), and the angle theta with the dipole axis (y-axis) is 45 deg. Dipole fields: E_r = (2P cos theta)/(4*pi*eps0 * r³) along r-direction, E_theta = (P sin theta)/(4*pi*eps0 * r³) along theta-direction. The Coulomb field from Q at origin acts along the line from O to A(a, a+b), at distance sqrt(a² + (a+b)²). For the total field to be zero, the dipole field and charge field must cancel. By symmetry analysis at theta = 45 deg, the dipole field is purely radial (since E_r and E_theta combine). The magnitude and direction conditions yield: the charge Q field must exactly oppose the dipole resultant at A. Working through the vector cancellation: distance from Q to A = sqrt(a² + (a+b)²). Setting the components equal leads to the condition b³ = 2*a³, giving b/a = 2^(1/3).

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