Question

At a point due to a point charge, the values of electric field intensity and potential are $32N{{C}^{-1}}$ and $16J{{C}^{-1}}$, respectively. Calculate the
(a) magnitude of the charge, and
(b) distance of the charge from the point of observation.

Answer 1

This problem can be solved by using the direct formula for the electric field intensity and the potential due to a point charge and solving them simultaneously to get the required values of the magnitude of the charge and the distance of the charge from the point of observation.

Formula Used:
$E=\dfrac{KQ}{{{r}^{2}}}$
$V=\dfrac{KQ}{r}$

Complete step-by-step answer :
Let us write the mathematical formula for the magnitude of the electric field intensity due to a point charge at a point.
The magnitude of electric field intensity $E$ due to a point charge of magnitude $Q$ at a distance $r$ away from it is given by
$E=\dfrac{KQ}{{{r}^{2}}}$ --(1)
Where $K=9\times {{10}^{9}}kg.{{m}^{3}}{{s}^{-2}}{{C}^{-2}}$ is the universal electric constant.
The magnitude of electric potential $V$ due to a point charge of magnitude $Q$ at a distance $r$ away from it is given by
$V=\dfrac{KQ}{r}$ --(2)
Now, let us analyze the question.
Let the magnitude of the charge be $Q$.
Let the distance of the point in question from this point charge be $r$.
The electric field at this point due to the point charge is given by $E=32N{{C}^{-1}}$.
The magnitude of electric potential at this point due to the point charge is given by $V=16J{{C}^{-1}}$
Now, using (1), we get
$32=\dfrac{KQ}{{{r}^{2}}}$
$\therefore KQ=32{{r}^{2}}$ --(3)
Now using (2), we get
$16=\dfrac{KQ}{r}$ --(4)
Putting (3) in (4), we get
$16=\dfrac{32{{r}^{2}}}{r}=32r$
$\therefore r=\dfrac{16}{32}=\dfrac{1}{2}=0.5m$ --(5)
Now, putting (5) in (3), we get
$KQ=32{{\left( 0.5 \right)}^{2}}=32\left( 0.25 \right)=8$
$\therefore Q=\dfrac{8}{K}=\dfrac{8}{9\times {{10}^{9}}}=8.89\times {{10}^{-10}}C$
Therefore, we have got the required values for the magnitude of the charge and the distance of the point from the point charge.

Note : Another way to solve this problem is to use the relation that the magnitude of the electric potential due to a point charge at a point is the product of the electric field at that point due to the point charge and the distance of the point from the point charge. By using this relation, we would have got the value of the distance in one step without substitution. However for calculating the magnitude of the charge we would have had to substitute the distance as we have done in the answer above.