Question

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

• A. $(1 / \sqrt{2}) a$
• B. $\sqrt{(3 / 2)} a$
• C. $\sqrt{(2 / 3)} a$
• D. $(1 / \sqrt{3}) a$

Answer 1

Answer: (C)

The equation of a semi-circular wave is
$x^{2}+y^{2} =a^{2} $ or $ y^{2}=a^{2}-x^{2} $
$I_{ rs } =\sqrt{\frac{1}{2 a} \int\limits_{-a}^{+a} y^{2} d x} $
$I_{ rms }^{2} =\frac{1}{2 a} \int\limits_{-a}^{+a}\left(a^{2}-x^{2}\right) d x$
$=\frac{1}{2 a} \int\limits_{-a}^{+a}\left(a^{2}-x^{2}\right) d x=\frac{1}{2 a}\left|a^{2} x-\frac{x^{3}}{3}\right|_{-a}^{+a} $
$=\frac{1}{2 a}\left(a^{3}-\frac{a^{3}}{3}+a^{3}-\frac{a^{3}}{3}\right)=\frac{2 a^{2}}{3} $
$I_{ rms } =\sqrt{\frac{2 a^{2}}{3}}=\sqrt{\frac{2}{3}} a$