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1. Electric Charges and Fields
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Consider a solid insulating sphere of radius $R$ with charge density varying as $\rho = \rho_0r^2$ ($\rho_0$ is a constant and r is measure from centre).Consider two points $A$ and $B$ at distance $x$ and $y$ respectively ($x < R, y > R$) from the centre. If magnitudes of electric fields at points $A$ and $B$ are equal, then
A
$x^2y = R^3 $
B
$x^3y^2 = R^5 $
C
$x^2y^3 = R^5 $
D
$\frac{x^4}{y} = R^5 $
Solution
$E_{x} \times 4 \pi x^{2}=\frac{\rho_{0} \int_{0}^{x} x^{2} \times 4 \pi x^{2} d x}{\in 0}$
$=\frac{4 \pi \rho_{0} x^{5}}{5 \in 0}$
$E_{y}=\frac{\rho_{0} \int_{0}^{R} x^{2} \times 4 \pi x^{2} d x}{\in_{0} \times 4 \pi y^{2}}=\frac{\rho_{0} \times R^{5}}{4 \in 0 y^{2}}$
$\frac{\rho_{0} x^{3}}{5 \in 0}=\frac{\rho_{0} R^{5}}{5 \in 0, y^{2}}$
$x^{3} y^{2}=R^{5}$
Standard 12
Physics
