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A circle, having radii 'a' has line char...

A circle, having radii 'a' has line charge distribution over its circumference having linear charge density `lambda = lambda_(0)cos^(2)theta`. Calculate the total electric charge residing on the circumference of the circle.
[Hint: `int_(0)^(2pi)cos^(2)d theta =pi`]

Text Solution

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The length of an infinitesimally small lint element shown in fig. is add, then the charge or the line element is

`dq = lambda ad theta (therefore lambda = q/l)`
`therefore dq = lambda_(0)cos^(2)theta.d theta`………(i)
`(therefore lambda = lambda_(0)cos^(2)theta)`
In order to calculate the total charge Q residing on the surface, we have to intigrate dq over the entire surface
`therefore Q = int_(0)^(2pi) lambda_(0)cos^(2)theta.d theta`
`=alambda_(0) int_(0)^(2pi)cos^(2)theta.d theta`
`therefore Q = alambda_(0)pi` `(therefore int_(0)^(2pi)cos^(2)theta d theta = pi)`
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