Question

Two charges q1 and q2 are placed at (0, 0, d) and (0, 0, -d) respectively. Find locus of points where the potential a zero.

NCERT Class XII
Physics - Exemplar Problems

Chapter 2. Potential and Capacitance

Answer 1

Potential at any point (x,y,z) in space due to these charges is
Due to q1
$\frac{k \times q1}{ \sqrt{ (x-0)^{2} + (y-0)^{2} + (z-d)^{2} } }$

Due to q2
$\frac{k \times q2}{ \sqrt{ (x-0)^{2} + (y-0)^{2} + (z+d)^{2} } }$

Now, according to the question , the sum has to be 0. So,
$\frac{k \times q1}{ \sqrt{ (x-0)^{2} + (y-0)^{2} + (z-d)^{2} } }$ + $\frac{k \times q2}{ \sqrt{ (x-0)^{2} + (y-0)^{2} + (z+d)^{2} } }$ = 0

Squaring both sides and simplifying,
$x^{2} ( q_1^{2} - q_2^{2} ) + y^{2} (q_1^{2} - q_2^{2}) + z^{2} (q_1^{2} - q_2^{2}) + \\ d^{2} (q_1^{2} - q_2^{2}) + 2 \times z \times d \times (q_1^{2} + q_2^{2})=0$