JEE MAIN - Physics (2016 - 9th April Morning Slot - No. 12)
The potential (in volts) of a charge distribution is given by.
V(z) = 30 $$-$$ 5x2 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.
V(z) = 30 $$-$$ 5x2 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.
$${\rho _0}$$ = 10 $${\varepsilon _0}$$ for $$\left| z \right|$$ $$ \le $$ 1 m and $${\rho _0} = 0$$ elsewhere
$${\rho _0}$$ = 20 $${\varepsilon _0}$$ in the entire region
$${\rho _0}$$ = 40 $${\varepsilon _0}$$ in the entire region
$${\rho _0}$$ = 20 $${\varepsilon _0}$$ for $$\left| z \right|$$ $$ \le $$ 1 m and $${\rho _0} = 0$$ elsewhere
Explanation
We know,
E(z) = $$-$$ $${{dv} \over {dz}}$$
$$ \therefore $$ E(z) = $$-$$ 10 z for $$\left| z \right| \le 1$$ m
and E(z) = 10 for $$\left| z \right| \ge 1$$ m
$$ \therefore $$ The source is an infinity large non conducting thick of thickness z = 2 m.
$$ \therefore $$ E = $${\sigma \over {2{\varepsilon _0}}}$$ = $${{\rho \left( 2 \right)} \over {2{\varepsilon _0}}}$$ = $${\rho \over {{\varepsilon _0}}}$$
$$ \therefore $$ $${\rho \over {{\varepsilon _0}}}$$ = 10
$$ \Rightarrow $$ $$\rho $$ = $${10\,{\varepsilon _0}}$$
E(z) = $$-$$ $${{dv} \over {dz}}$$
$$ \therefore $$ E(z) = $$-$$ 10 z for $$\left| z \right| \le 1$$ m
and E(z) = 10 for $$\left| z \right| \ge 1$$ m
$$ \therefore $$ The source is an infinity large non conducting thick of thickness z = 2 m.
$$ \therefore $$ E = $${\sigma \over {2{\varepsilon _0}}}$$ = $${{\rho \left( 2 \right)} \over {2{\varepsilon _0}}}$$ = $${\rho \over {{\varepsilon _0}}}$$
$$ \therefore $$ $${\rho \over {{\varepsilon _0}}}$$ = 10
$$ \Rightarrow $$ $$\rho $$ = $${10\,{\varepsilon _0}}$$
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