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

The correct option is C $E_1\left(r_0/2\right)=2E_2\left(r_0/2\right)$

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