Question

A regular hexagon of side $10cm$ has a charge $5\mu C$ at each of its vertices. Calculate potential at the centre of the hexagon.

Answer 1

Hint: Calculate the electric potential at the centre of the hexagon due to the particle on one of its vertices. The total potential can be then the potential at one of the vertices multiplied by the total number of vertices. As the vertices are not intersecting, the electric potential can be calculated this way.
Formula used:
$E=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\times \dfrac{q}{d}$

Complete answer:
Let us assume the length of each vertex from the centre as $d=10cm$. Also, given in the question that the length of each side is $10cm$. The charge is given as $q=5\times {{10}^{-6}}C$.now, let us calculate the electric potential at the point, let us say, a;
$V=\dfrac{q}{4\pi {{\varepsilon }_{0}}d}$
For all the six vertices, the electric potential will be,
$V=\dfrac{6q}{4\pi {{\varepsilon }_{0}}d}$
The value of $\dfrac{1}{4\pi {{\varepsilon }_{0}}}=9\times {{10}^{9}}N{{C}^{-2}}{{m}^{-2}}$where ${{\varepsilon }_{0}}$is the permittivity of free space.
Now,
$\begin{align}
  & V=\dfrac{9\times {{10}^{9}}\times 5\times {{10}^{-6}}\times 6}{0.1} \\
 & V=2.7\times {{10}^{6}}V \\
\end{align}$
Therefore, the potential at the centre of the hexagon due to all the vertices is $2.7\times {{10}^{6}}V$.

Additional Information:
The electric potential energy for a positive charge will increase as it goes against an electric field and decreases when it goes along with the electric field. The negative charge will experience the opposite. The electric potential is the potential energy per unit charge. The concept of potential is used to express the effect of the electric field of any source in terms of the region present inside the field. The electric field charges will flow from one place to another because of the difference in electric potential at the two places.

Note:
The electric potential only has magnitude but no direction. It is hence a scalar quantity. Both electric force and electric field are vectors only. If the electric field at any point is zero, it doesn’t mean that electric potential is also zero at that point. Electric potential is zero when the charges are equal and opposite and are at same distance from a fixed point.