Question

Two infinite sheets having charge densities ${\sigma }_1$ and ${\sigma }_2$ are placed in two perpendicular planes whose two-dimensional view is shown in the figure. Net electric field at point $\text{P}$ is

• A. $\frac{\sqrt{{\sigma }_1^2+{\sigma }_2^2}}{2{ϵ}_0}$
• B. $\frac{{\sigma }_1^2+{\sigma }_2^2}{{ϵ}_0}$
• C. $\frac{{\sigma }_1^2-{\sigma }_2^2}{2{ϵ}_0}$
• D. Data insufficient

Answer 1

The correct option is A $\frac{\sqrt{{\sigma }_1^2+{\sigma }_2^2}}{2{ϵ}_0}$

Electric field at point $\text{P}$ is given in the following figure


Electric field at point $\text{P}-$

Due to sheet 1, $E_1=\frac{{\sigma }_1}{2{ϵ}_0}$

Due to sheet 2, $E_2=\frac{{\sigma }_2}{2{ϵ}_0}$

So, net electric field at point $\text{P}$,

$E_{net}=\sqrt{E_1^2+E_2^2}$

$\Rightarrow E_{net}=\sqrt{{\left(\frac{{\sigma }_1}{2{ϵ}_0}\right)}^2+{\left(\frac{{\sigma }_2}{2{ϵ}_0}\right)}^2}=\frac{\sqrt{{\sigma }_1^2+{\sigma }_2^2}}{2{ϵ}_0}$

So, option $\left(a\right)$ is correct.