Question

The potential (in volts) of a charge distribution is given by
$V(z)=30-5z^2$ for $|z|\le 1m$
$V(z)=35-10|z|$ for $|z|\ge 1m$.
$V(z)$ does not depend on $x$ and $y$. If this potential is generated by a constant charge per unit volume ${\rho }_0$ (in units of ${ϵ}_0)$ which is pread over a certain region, then choose the correct statement.

• A. ${\rho }_0=10{ϵ}_0$ for $|z|\le 1m$ and ${\rho }_0=0$ elsewhere
• B. ${\rho }_0=20{ϵ}_0$ in the entire region
• C. $rh{o}_0=40{ϵ}_0$ in the entire region
• D. ${\rho }_0=20{ϵ}_0$ for $|z|\le 1m$ 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.
$\therefore 10Z\cdot 10A=\frac{\rho \cdot A\propto Z}{{\epsilon }_0}$
${\rho }_0=10{\epsilon }_{0}for\left|z\right|\le 1m$