Determine the rms value of a semi-circular current wave which has a maximum value of a.

Options

A. `(1//sqrt(2))a`
B. `sqrt((3//2))a`
C. `sqrt((2//3))a`
D. `(1//sqrt(3))a`

Answer

C

Explanation

the equation of a semi-circular wave is
`x^(2)+y^(2)= a^(2) or y^(2) =a^(2)-x^(2)`
`(I_(rms)) = sqrt((1)/(2a) int_(-a)^(+a) y^(2)dx)`
`=I_(rms)^(2) = (1)/(2a) int _(-a)^(+a) (a^(2)-x^(2))dx`
`=1/(2a) int_(-a)^(+a) (a^(2)-x^(2))dx=1/(2a)|a^(2)x-(x^3)/(3)|_(-a)^(+a)`
`(1)/(2a)(a^(3)=(a^3)/(3)+a^(3)-(a^3)/(3))=(2a^(3))/(3)`
`I_(rms) = sqrt((2a^(2))/(3)) = sqrt((2)/(3)) a`.