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The linear charge density on a dielectri...

The linear charge density on a dielectric ring of radius `R` vanes with `theta` as `lambda = lambda_0 cos theta//2`, where `lambda_0` is constant. Find the potential at the center `O` of the ring [in volt].

A

16

B

0

C

5

D

10

Text Solution

Verified by Experts

The correct Answer is:
B
The charge on the infinitesimal elements of arc that subtends an angle `d theta` at the center of the ring is
`d Q = lambda R d theta = lambda_0 " cos " theta/(2) Rd theta`
Potential at the center of ring due to charge `d Q` is
`d V = (1)/(4 pi epsilon_0)(dQ)/(R) = (lambda_0 " cos " theta/2 Rd theta)/(4 pi epsilon_0 R)`
or `V = int d V = (lambda_0)/(4 pi epsilon_0) underset(0) overset(2 pi)(int) " cos " theta/(2) d theta`
=`(lambda_0)/(4 pi epsilon_0)[ (" sin " theta/(2))/(1/2)]_0^(2 pi) = 0 V`.
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