A point particle of charge Q is located at P along the axis of an electric dipole 1 at a distance r as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r. The dipoles are made of opposite charge q separated by a distance 2 a. For the charge particle at P not to experience any net force, which of the following correctly describes the situation?

Solution

Physics sir · 3 steps

Use the exact dipole field formulas, not the far-field ones, and set the two magnitudes equal.

1. Axial field. On the axis of dipole 1, the charges +q and -q sit at distances r-a and r+a from P. The field is kq[1/(r-a)²-1/(r+a)²] = 4kqar/(r²-a²)².
2. Equatorial field. P is at distance r on the perpendicular bisector of dipole 2, so each charge is √(r²+a²) away. Their components along the dipole add to 2kqa/(r²+a²)^(3/2). This field points opposite to the axial field at P, so the two can cancel.

   4kqar/(r²-a²)² = 2kqa/(r²+a²)^(3/2) ⇒ (1-x²)² = 2(1+x²)^(3/2), x = a/r

   x = a/r	(1-x²)²	2(1+x²)^(3/2)	Balanced?
   0.5	0.56	2.8	no
   3	64	63.2	yes, almost equal
   10	9801	2030	no
   20	159201	16060	no

3. So. The two field magnitudes match at a/r ≈ 3 (the exact root is about 2.98), which is option (b). Strictly, with a > r the point P lies between the charges of dipole 1, but the question intends this magnitude match and the options are built on it.

The idea to keep. When r and a are comparable, do not use 2kp/r³ or kp/r³. Write the exact fields, divide through by a power of r, and test the options as values of a/r.

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