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The potential (in volts ) of a charge distribution is given by
$V(z)\, = \,30 - 5{z^2}for\,\left| z \right| \le 1\,m$
$V(z)\, = \,35 - 10\,\left| z \right|for\,\left| z \right| \ge 1\,m$
$V(z)$ does not depend on $x$ and $y.$ If this potential is generated by a constant charge per unit volume $\rho _0$ (in units of $\varepsilon _0$ ) which is spread over a certain region, then choose the correct statement
${\rho _0}\, = \,20\,{\varepsilon _0}$ in the entire region
${\rho _0}\, = \,10\,{\varepsilon _0}$ for $\left| z \right|\, \le 1\,\,m$ and $P_0 = 0$ elsewhere
${\rho _0}\, = \,20\,{\varepsilon _0}$ for $\left| z \right|\, \le 1\,\,m$ and $P_0 = 0$ elsewhere
${\rho _0}\, = \,40\,{\varepsilon _0}$ in the entire region
Solution
$\Sigma_{1}=\frac{-d v}{d r}=10|z|$
$\Sigma_{2}=\frac{-\mathrm{dv}}{\mathrm{dr}}=10 \quad(\text { constant : } \mathrm{E})$
$\therefore $ The source is an infinity large non conducting thick plate of thickness $2\, \mathrm{m}$.
$\therefore 10 \mathrm{Z} \cdot 10 \mathrm{A}=\frac{\rho \cdot \mathrm{A} \propto \mathrm{Z}}{\varepsilon_{0}}$
$\rho_{0}=10 \mathrm{e}_{0}$ for $|\mathrm{z}| \leq 1\, \mathrm{m}$
