An annular disc has inner and outer radius R1 and R2 respectively. Charge is uniformly distributed. Surface charge density is σ. Find the electric field at any point distant y along the axis of the disc.
A
σy2ϵ0⎡⎢
⎢⎣1√R21+y2−1√R22+y2⎤⎥
⎥⎦
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B
σ2ϵ0
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C
σy2ϵ0(R2−R1)
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D
σ2ϵ0logR2+yR1+y
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Solution
The correct option is Aσy2ϵ0⎡⎢
⎢⎣1√R21+y2−1√R22+y2⎤⎥
⎥⎦
Consider a hypothetical ring of radius x and thickness dx on the disc.
The charge on the hypothetical ring : dq=σ.2πxdx
Now the electric field at point P due to the ring is :
dE=dq4πϵ0×y(x2+y2)3/2
dE=σ.2πxdx4πϵ0×y(x2+y2)3/2
For complete annular disc, E=∫dE=σy2ϵ0∫x(x2+y2)3/2dx