Question:
An infinite line charge produces a field of $9 \times 10^{4} \mathrm{~N} / \mathrm{C}$ at a distance of $2 \mathrm{~cm}$. Calculate the linear charge density.
Solution:
Electric field produced by the infinite line charges at a distance d having linear charge density λ is given by the relation,
$E=\frac{\lambda}{2 \pi \in_{0} d}$
$\lambda=2 \pi \in_{0} d E$
Where,
d = 2 cm = 0.02 m
$E=9 \times 10^{4} \mathrm{~N} / \mathrm{C}$
$\epsilon_{0}=$ Permittivity of free space
$\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{~N} \mathrm{~m}^{2} \mathrm{C}^{-2}$
$\lambda=\frac{0.02 \times 9 \times 10^{4}}{2 \times 9 \times 10^{9}}$
$=10 \mu \mathrm{C} / \mathrm{m}$
Therefore, the linear charge density is $10 \mu \mathrm{C} / \mathrm{m}$.