Question
A point charge of magnitude $$ + 1\mu C$$ is fixed at $$\left( {0,0,0} \right).$$ An isolated uncharged spherical conductor, is fixed with its center at $$\left( {4,0,0} \right).$$ The potential and the induced electric field at the centre of the sphere is :
A.
$$1.8 \times {10^5}\,V$$ and $$ - 5.625 \times {10^6}\,V/m$$
B.
$$0\,V$$ and $$0\,V/m$$
C.
$$2.25 \times {10^5}V$$ and $$ - 5.625 \times {10^6}\,V/m$$
D.
$$2.25 \times {10^5}V$$ and $$0\,V/m$$
Answer :
$$2.25 \times {10^5}V$$ and $$ - 5.625 \times {10^6}\,V/m$$
Solution :
$$q = 1\,\mu C = 1 \times {10^{ - 6}}C,r = 4\,cm,4 \times {10^{ - 2}}m$$
Potential $$V = \frac{{kq}}{r} = \frac{{9 \times {{10}^9} \times {{10}^{ - 6}}}}{{4 \times {{10}^{ - 2}}}}$$
$$ = 2.25 \times {10^5}\,V.$$
Induced electric field $$E = - \frac{{kq}}{{{r^2}}}$$
$$ = \frac{{9 \times {{10}^9} \times 1 \times {{10}^{ - 6}}}}{{16 \times {{10}^{ - 4}}}} = - 5.625 \times {10^6}\,V/m$$