The linear charge density of a uniform semicircular wire varies with `theta` shown in the figure as `lambda=lambda_(0)costheta` if `R` is the radius of the loop ,of the charge on the wire is

Options

A. `lambda_(0)R`
B. `3lambda_(0)R`
C. `(2lambda_(0)R)/(3)`
D. `2lambda_(0)R`

Answer

D

Explanation

`rArr int_(0)^(1)lambda_(0)x^(2)(dx) rArr q=(lambda_(0)l^(3))/(3)`

`dQ=lambdaR(dtheta)=lambda_(0)costhetaRdtheta`
`int_(-pi//2)^(pi//2) lambda_(0)Rcostheta d theta `
`=lambda_(0)R(sintheta)_(-pi//2)^(pi//2)=2lambda_(0)R`