Question

Find out the surface charge density at the intersection of point $x=3m$plane and $x-$axis in the region of uniform linear charge density of $8nC/m$ lying along the $z-$axis in free space.

• A. $47.88C/m$
• B. $0.07nC{m}^{-2}$
• C. $0.424nC{m}^{-2}$
• D. $4.0nC{m}^{-2}$

Answer 1

The correct option is C

$0.424nC{m}^{-2}$

Step 1. Given data.

Intersection point, $x=3m$,

Uniform line charge that is linear charge density, $\lambda$$=8nC/m$,

Assume coulomb constant $k=\frac{1}{4\pi {\epsilon }_0}$ (where ${\epsilon }_0$ is the permittivity of free space),

${\epsilon }_0=8.85\times {10}^{-12}F{m}^{-1}$ and $\pi =3.14$.

Step 2. The formula for Electric field due to wire,

Electric field due to wire is given by $E=\frac{2k\lambda }{x}\cdots \left(1\right)$,

Where, $E$ is the Electric field due to wire, $k$ is the coulomb constant, $\lambda$ is the linear charge density and $x$ is the perpendicular distance of the point from the line charge.

Step 3. The formula for Electric field with surface charge density,

The electric field with surface charge density $E=\frac{\sigma }{{\epsilon }_{\circ }}\dots \left(2\right)$,

Where, $E$ is the electric field with surface charge density, $\sigma$ is the surface charge density and ${\epsilon }_0$ is the permittivity of free space,

Step 4. Calculating surface charge density,

Now, by equating the above equation $\left(1\right)$ and equation $\left(2\right)$ we get,

$$\begin{gathered} \Rightarrow \frac{2k\lambda }{x}=\frac{\sigma }{{\epsilon }_{\circ }} \\ \Rightarrow 2\frac{1}{4\pi {\epsilon }_{\circ }}\frac{\lambda }{x}=\frac{\sigma }{{\epsilon }_{\circ }} \\ \Rightarrow \sigma =\frac{2\lambda {\epsilon }_0}{4\pi x{\epsilon }_{\circ }} \\ \Rightarrow \sigma =\frac{\lambda }{2\pi x} \\ \Rightarrow \sigma =\frac{8\times {10}^{-9}}{2\times 3.14\times 3}C{m}^{-2} \\ \Rightarrow \sigma =\frac{8\times {10}^{-9}}{18.84}C{m}^{-2} \\ \Rightarrow \sigma =0.424\times {10}^{-9}C{m}^{-2} \\ \Rightarrow \sigma =0.424nC{m}^{-2} \end{gathered}$$

Therefore, the surface charge density, $\sigma$ is$0.424nC{m}^{-2}$.

Hence, the correct option is C.