Question

Let $E_1(r),E_2(r)$ and $E_3(r)$ be the respective electric fields at a distance $r$ from a point charge $Q,an$ infinitely long wire with constant linear charge density $\lambda$, and an infinite plane with uniform surface charge density $\sigma$. If $E_1\left(r_0\right)=E_2\left(r_0\right)=E_3\left(r_0\right)$ at a given distance $r_0$, then

• A. $Q=4\sigma \pi r_0^2$
• B. $r_0=\frac{\lambda }{2\pi \sigma }$
• C. $E_1\left(r_0/2\right)=2E_2\left(r_0/2\right)$
• D. $E_2\left(r_0/2\right)=4E_3\left(r_0/2\right)$

Answer 1

Answer: (C)

$\frac{Q}{4\pi {\epsilon }_0{r}_0^2}=\frac{\lambda }{2\pi {\epsilon }_0{I}_0}=\frac{\sigma }{2{\epsilon }_0}$
$E_1\left(\frac{r_0}{2}\right)=\frac{Q}{\pi {\epsilon }_0{I}_0^2},$
$E_2\left(\frac{r_0}{2}\right)=\frac{\lambda }{\pi {\epsilon }_0{I}_0},$
$E_3\left(\frac{r_0}{2}\right)=\frac{\sigma }{2{\epsilon }_0}$
$\therefore E_1\left(\frac{r_0}{2}\right)$
$=2E_2\left(\frac{r_0}{2}\right)$