Let ${\mathrm{E}}_1\left(\mathrm{r}\right),\mathrm{ }{\mathrm{E}}_2\left(\mathrm{r}\right)$ and ${\mathrm{E}}_3\left(\mathrm{r}\right)$ be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density $\mathrm{\lambda }$ , and an infinite plane with uniform surface charge density $\mathrm{\sigma }$ . If ${\mathrm{E}}_1\left({\mathrm{r}}_0\right)={\mathrm{E}}_2\left({\mathrm{r}}_0\right)={\mathrm{E}}_3\left({\mathrm{r}}_0\right)$ at a given distance ${\mathrm{r}}_0$ , then

• A

  $\mathrm{Q}=4\mathrm{\sigma }\mathrm{\pi }{\mathrm{r}}_0^2$
• B

  ${\mathrm{r}}_0=\frac{\mathrm{\lambda }}{2\mathrm{\pi }\mathrm{\sigma }}$
• Correct answer
  C

  ${\mathrm{E}}_1\left(\frac{{\mathrm{r}}_0}{2}\right)=2{\mathrm{E}}_2\left(\frac{{\mathrm{r}}_0}{2}\right)$
• D

  ${\mathrm{E}}_2\left(\frac{{\mathrm{r}}_0}{2}\right)=4{\mathrm{E}}_3\left(\frac{{\mathrm{r}}_0}{2}\right)$

Hence, the correct answer is option: 'C'.