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

Options

A. 2.515 a
B. 1.815 a
C. 0.615 a
D. 0.816 a

Answer

D

Explanation

The equation of a semi-circular wave (shown in figure) 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))`
or `I_("rms")^(2) = (1)/(2a) int_(-a)^(+a) (a^(2) - x^(2)) dx`
`= (1)/(2a) int_(-a)^(+a) (a^(2) dx - x^(2) dx) =(1)/(2a) |a^(2) x - (x^(2))/(3)|_(-a)^(+a)`
`= (1)/(2a) (a^(2) - (a^(3))/(3) + a^(3) - (a^(3))/(3)) = (2a^(2))/(3)`
`:. I_("rms") = sqrt(2a^(2)//3) = 0.81 6a`.