An infinitely long thin wire, having a uniform charge density per unit length of $5 nC / m$, is passing through a spherical shell of radius $1 m$, as shown in the figure. A $10 nC$ charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points $P$ and $R$, in Volt, is. . . .
[Given: In SI units $\frac{1}{4 \pi \epsilon_0}=9 \times 10^9, \ln 2=0.7$. Ignore the area pierced by the wire.]
IIT 2024, Medium
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due to wire
$dV =-\overrightarrow{ E } \cdot \overrightarrow{ dx }$
$\int_{T_{ r }}^{ V _{ R }} dV =-\int_{0.5}^2 \frac{2 k \lambda}{ x } dx$
$V _{ R }- V _{ P }=-2 k \lambda \ln \frac{2}{0.5}$
$=-2 \times 9 \times 10^9 \times 3 \times 10^{-9} \times 2 \times 0.7=-126$
due to sphere
$v_R-v_P=\frac{k Q}{2}-\frac{k Q}{1}=-\frac{k Q}{2}=\frac{-9 \times 10^9 \times 10 \times 10^{-9}}{2}$
$=-45 V$
$v_R-v_P=-126-45=-171 V$
$v_p-v_R=171 V$
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