Question

A +3.0 nC charge is at x = 0 cm and a -1.0 nC charge is at x = 4 cm. At what point or points on the x-axis is the electric potential zero?

Short answer

The points are 3 cm and 6 cm

Step-by-step solution

Definition of electric potential 

Assume for $q_1$, x=0 and for $q_2$, x=+4.0 cm

When $x<0$positive charges> negative charge. due to this reason, they can not be cancelled.

At when$0<x<4$

localid="1648130310708" $$\begin{gathered} V=1/\left(4\pi {\epsilon }_0\right)(\frac{q_1}{r_0}+\frac{q_2}{r_4}) \\ {q}_1=3\times {10}^{-9}C \\ {q}_2=1\times {10}^{-9}C \\ {r}_0=x \\ {r}_4=4-x \\ V=1/\left(4\pi {\epsilon }_0\right)(\frac{3\times {10}^{-9}C}{x}+\frac{1\times {10}^{-9}C}{4-x}) \\ AtV=0 \\ x=3.0cm \end{gathered}$$

[Note: the value of electric potential for two charges is 0, V=0 V]

Definition of electric potential 

At when x> 4

$\begin{array}{rcl}V& =& 1/\left(4\pi {\epsilon }_0\right)(\frac{q_1}{r_0}+\frac{q_2}{r_4})\\ q_1& =& 3\times {10}^{-9}C\\ q_2& =& 1\times {10}^{-9}C\\ r_0& =& x\\ r_4& =& x-4\\ V& =& 1/\left(4\pi {\epsilon }_0\right)(\frac{3\times {10}^{-9}C}{x}+\frac{1\times {10}^{-9}C}{x-4})\\ AtV& =& 0\\ x& =& 6.0cm\end{array}$

[Note: the value of electric potential for two charges is 0, V=0 V]