The potential (in volts) of a charge distribution is given by

$\mathrm{V}\left(\mathrm{z}\right)=30-5{\mathrm{z}}^2$ for $\left|\mathrm{z}\right|\le 1\mathrm{ }\mathrm{m}$

$\mathrm{V}\left(\mathrm{z}\right)=35-10\mathrm{ }\left|\mathrm{z}\right|$ for $\left|\mathrm{z}\right|\ge 1\mathrm{ }\mathrm{m}$ .

$\mathrm{V}\left(\mathrm{z}\right)$ does not depend on x and y. If this potential is generated by a constant charge per unit volume ${\mathrm{\rho }}_0$ (in units of ${\mathrm{ϵ}}_0$ ) which is spread over a certain region, then choose the correct statement.

• A

  ${\mathrm{\rho }}_0=20\mathrm{ }{\mathrm{ϵ}}_0$ in the entire region
• Correct answer
  B

  ${\mathrm{\rho }}_0=10\mathrm{ }{\mathrm{ϵ}}_0$ for $\left|\mathrm{z}\right|\le 1\mathrm{ }\mathrm{m}$ and ${\mathrm{\rho }}_0=0$ else where
• C

  ${\mathrm{\rho }}_0=20\mathrm{ }{\mathrm{ϵ}}_0$ for $\left|\mathrm{z}\right|\le 1\mathrm{ }\mathrm{m}$ and ${\mathrm{\rho }}_0=0$ else where
• D

  ${\mathrm{\rho }}_0=40\mathrm{ }{\mathrm{ϵ}}_0$ in the entire region

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