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

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Answer & Solution

Correct Answer

(B) \({\rho _0}\, = \,10\,{\varepsilon _0}\) for \(\left| z \right|\, \le 1\,\,m\) and \(P_0 = 0\) elsewhere

Detailed explanation

\(\Sigma_{1}=\frac{-d v}{d r}=10|z|\)

\(\Sigma_{2}=\frac{-\mathrm{dv}}{\mathrm{dr}}=10 \quad(\text { constant : } \mathrm{E})\)

\(\therefore \) 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 \(|\mathrm{z}| \leq 1\, \mathrm{m}\)

Same subject

From JEE Mains

List \(I\)	List \(II\)
\(A\) Troposphere	\(I\) Approximate \(\quad 65-75\, km\) over Earth's surface
\(B\) E-Part of Stratosphere	\(II\) Approximate \(300\,km\) over Earth's surface
\(C\) \(F _2\)-Part of Thermosphere	\(III\) Approximate \(10\,km\) over Earth's surface
\(D\) \(F _2\)-Part of Thermosphere	\(IV\) Approximate \(100\,km\) over Earth's surface