A conducting sphere of radius 10 cm has an unknown charge. If the electric field, 20 cm from the centre of the

Question

A conducting sphere of radius 10 cm has an unknown charge. If the electric field, 20 cm from the centre of the sphere is $1.5 	imes 10^3$ N/C and points radially inward, what is the net charge on the sphere?

Accepted Answer

The electric field $E$ at a distance $r$ from the centre of a charged sphere (where $r > R$) is given by $E = \frac{k|q|}{r^2}$. 
Given:
$E = 1.5 	imes 10^3$ N/C
$r = 20$ cm $= 0.2$ m
$k = 9 	imes 10^9$ N m$^2$/C$^2$

Solving for $|q|$:
$|q| = \frac{E r^2}{k} = \frac{(1.5 	imes 10^3) (0.2)^2}{9 	imes 10^9} = \frac{1.5 	imes 0.04 	imes 10^3}{9 	imes 10^9} = \frac{0.06 	imes 10^{-6}}{9} 	imes 10^9$ (wait, arithmetic check: $0.06 	imes 10^3 / 9 	imes 10^9 = 6.67 	imes 10^{-9}$)
$|q| \approx 6.67 	imes 10^{-9}$ C $= 6.67$ nC.
Since the electric field points radially inward, the charge must be negative. Thus, $q = -6.67$ nC. (See NCERT Physics Class 12, Exercise 1.20).

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