Question

The potential (in volts) of a charge distribution is given by
$V(z) = 30 - 5z^2 $ for $|z| \leq 1 \, m $
$V(z) = 35 - 10 |z|$ for $|z| \geq \, 1 \, m$.
$V(z)$ does not depend on $x$ and $y$. If this potential is generated by a constant charge per unit volume $\rho_0$ (in units of $\epsilon_0 )$ which is pread over a certain region, then choose the correct statement.

• A. $\rho_0 = 10 \, \epsilon_0$ for $|z| \leq 1 \, m$ and $\rho_0 = 0$ elsewhere
• B. $\rho_0 = 20 \, \epsilon_0$ in the entire region
• C. $rho_0 = 40 \, \epsilon_0$ in the entire region
• D. $\rho_0 = 20 \, \epsilon_0$ for $|z| \leq 1 \, m$ and $\rho_0 = 0$ elsewhere

Answer 1

Answer: (A)

$\sum_{1}=\frac{-dv}{dr}=10\left|z\right|$
$\sum_{2}=-\frac{dv}{dr}=10\left(constant: E\right)$
$\therefore $ The source is an infinity large non conducting thick plate of thickness 2m.
$\therefore 10Z\cdot10A=\frac{\rho\cdot A\propto Z}{\varepsilon_{0}}$
$\rho_{0}=10\varepsilon_{0}\,for\,\left|z\right|\le1m$