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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.

Physics - Exemplar Problems

Chapter 2. Potential and Capacitance



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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

0 0 0
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