Question

Six point charges are kept at the vertices of a regular hexagon of side $L$ and centre $O$, as shown in the figure. Give that $K=\frac{1}{4\pi {\epsilon }_0}\frac{q}{L^2}$, which of the following statement(s) is (are) correct?

• A. The electric field at O is 6 K along OD.
• B. The potential at O is zero.
• C. The potential at all points on the line PR is same.
• D. The potential at all points on the line ST is same.

Answer 1

Answer: (A), (B), (C)

It can be seen, from the figure depicted here, that $\left|{\vec{E}}_{34}\right|=\left|{\vec{E}}_{12}\right|=2K$.
Therefore, ${\vec{E}}_{34}+{\vec{E}}_{12}={\vec{E}}^{\prime }=2K$ is along $OD$.
Also the net field $\vec{E}={\vec{E}}^{\prime }+{\vec{E}}_{56}$
$=2K+4K=6K$ is along $OD$.

The three sets of the dipoles create a zero potential at the centre.
The vertical line $PR$ is along the equatorial axis of three sets of dipoles.
Now, the potential at all points along the line $PR$ is zero.
The line $ST$ lies along the axis of the dipole $(+2q,-2q)$ and the potential varies from point to point.