Question

Three charges $+2q,-q$ and $-q$ are located at the vertices of an equilateral triangle. At the center ${}^{\prime }{O}^{\prime }$ of the triangle,

• A. the field is zero but potential is non-zero.
• B. the field is non-zero but potential is zero.
• C. both field and potential are zero.
• D. both field and potential are non-zero.

Answer 1

The correct option is B the field is non-zero but potential is zero.


Potential due to the system of charges at point ${}^{\prime }{O}^{\prime }$ which is equidistant $\left(r\right)$ from each charge is given by
$$V=\frac{1}{4\pi {\epsilon }_0}\sum _{i=1}^3\frac{Q_i}{r_i}$$
$\Rightarrow V=\frac{1}{4\pi {\epsilon }_{0}r}\sum _{i=1}^3{Q}_i$

From the data given in the question

$\sum _{i=1}^3{Q}_i=+2q-q-q=0$

$\Rightarrow V=0$

Let us consider a unit test charge placed at ${}^{\prime }{O}^{\prime }$

The electric field acting on the test charge is as shown in the figure below.


Since, Downward component of electric field does not cancel we can conclude that electric field at point ${}^{\prime }{O}^{\prime }$ is non-zero.

Alternate method to show Electric field is Non-zero:

Since, $V=0$ we can say that monopole moment is equal to zero. When monopole moment (Total charge) is zero, the given system of charges can be considered as two dipoles as shown below.



From the figure, we can deduce that, the point ${}^{\prime }{O}^{\prime }$ is along the equatorial line of an electric dipole.

Since, direction of electric field is opposite to the direction of dipole moment.


So, we can say that, Electric field due to two dipoles at ${}^{\prime }{O}^{\prime }$ is non-zero.

Hence, option (b) is the correct answer.