Question

An infinite line charge produces a field of $9 \times {10^4}N$/ $C$ at a distance of $2$ cm. Calculate the linear charge density.
A. $0.1\,\mu C{m^{ - 1}}$
B. $0.01\,\mu C{m^{ - 1}}$
C. $10\,\mu C{m^{ - 1}}$
D. $1\,\mu C{m^{ - 1}}$

Answer 1

The electric force per unit charge is known as the electric field. Gauss' law can be used to calculate the electric field of an infinite line charge with a uniform linear charge density. The electric field has the same magnitude at any point of the cylinder and is oriented outward when considering a Gaussian surface in the shape of a cylinder with radius $r$.

Complete step by step answer:
The Electric field produced by the infinite line charges at a distance represented as d, having linear charge density as $\lambda$is given by the relation:
$E = \dfrac{\lambda }{{2\pi {\varepsilon _0}d}}$
Here we need to find the linear charge density which is $\lambda$. So, to find the value of $\lambda$, the above equation can be rewritten as:
$\lambda = (2\pi {\varepsilon _0}d) \times E$
From the question we know that,
$d = 2cm = 0.02m$
$\Rightarrow E = 9 \times {10^4}N$/$C$
${\varepsilon _0}$ is the permittivity of free space.
We also know that ,
$\dfrac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}N{m^2}{C^{ - 2}}$
$\Rightarrow \dfrac{1}{{2\pi {\varepsilon _0}}} = 2 \times 9 \times {10^9}N{m^2}{C^{ - 2}}$
Substituting all these values in the equation,
$\lambda = \dfrac{{0.02 \times 9 \times {{10}^4}}}{{2 \times 9 \times {{10}^9}}}$
$\therefore \lambda= 10\,\mu C$/$m$

Hence, the right answer is the option C.

Note: To find the electric field of an infinite line charge with a uniform linear charge density the Gauss’ Law is applied by using an appropriate gaussian surface. The electric field is proportional to the linear charge density and inversely proportional to the distance from the line.