Question

Let $E_1\left(r\right),E_2\left(r\right)$ and $E_3\left(r\right)$ be the respective electric fields at a distance $r$ from a point charge $Q$, an infinitely long wire 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 $E_1\left(r_0/2\right)={2E}_2\left(r_0/2\right)$

Given

$E_1\left(r_0\right)=E_2\left(r_0\right)=E_3\left(r_0\right)$

$\frac{Q}{4\pi a_o{r}_o^2}=\frac{\lambda }{2\pi a_{0}r}=\frac{6}{2a_0}$

$E_1\left(\frac{r_0}{2}\right)=\frac{Q}{4\pi a_0{\left(\frac{r_0}{2}\right)}^2}=2E_2\left(r_0\right)$

$\frac{E_1\left(\frac{r_0}{2}\right)}{E_2\left(\frac{r_0}{2}\right)}=\frac{4E_1\left(r_0\right)}{2E_2\left(r_0\right)}$

$E_1\left(\frac{r_0}{2}\right)=2E_2\left(\frac{r_0}{2}\right)$