Question

The electric field strength $10.0cm$from a very long charged wire is $2000N/C$. What is the electric field strength $5.0cm$ from the wire?

Short answer

The electric field strength at $5cm$from the wire is$1000N/C$.

Step-by-step solution

Given information 

Electric field strength due to a charged wire at$10cm\text{is}2000N/C$.

Explanation

Electric field due to the wire at $r_1=10cm$is $2000N/C$

$E_1=\frac{\lambda }{2\pi {\epsilon }_0{r}_1}\dots \dots \left(1\right)$

Electric field due to the long charged wire at $r_2=5cm$is

$E_2=\frac{\lambda }{2\pi {\epsilon }_0{r}_2}\dots \dots \left(2\right)$

Divide equation $\left(1\right)\text{and}\left(2\right)$

$$\begin{gathered} \frac{E_2}{E_1}=\frac{\frac{\lambda }{2\pi {\epsilon }_0{r}_2}}{\frac{\lambda }{2\pi {\epsilon }_0{r}_1}} \\ \frac{E_2}{E_1}=\frac{r_1}{r_2} \\ {E}_2=E_1\left(\frac{r_1}{r_2}\right) \\ =2000N/C\left(\frac{5cm}{10cm}\right) \\ =1000N/C \end{gathered}$$