question_answer

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

\[V(z)=30-5{{z}^{2}}\]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 \[{{\varepsilon }_{0}}\]) which is spread over certain region ,then choose the correct statement. [JEE ONLINE 09-04-2016]

A) \[{{\rho }_{0}}=40{{\varepsilon }_{0}}\]in the entire region

B) \[{{\rho }_{0}}=20{{\varepsilon }_{0}}\] in the entire region

C) \[{{\rho }_{0}}=20{{\varepsilon }_{0}}\] for \[|z|\]in\[\le 1m\]and \[{{\rho }_{0}}=0\]else where

D) (d\[{{\rho }_{0}}=10{{\varepsilon }_{0}}\]for \[|z|\]in \[\le 1m\]and\[{{\rho }_{0}}=0\] else where

Correct Answer: D

Solution :

[d] \[|E=\left\{ \begin{matrix}    10zv/m & |z|\le 1m  \\    10v/m & |z|\ge 1m  \\ \end{matrix} \right.\]

If should be for a sheet lying in x.y plane of thickness z = 2m.

For \[|z|\ge 1mE=\frac{\sigma }{2{{\in }_{0}}}=\frac{\rho (2)}{2{{\in }_{0}}}=\frac{\rho }{{{\in }_{0}}}=10\Rightarrow P=10{{\in }_{0}}\]

z = .1 z=0 z = +1