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 $y^2=a^2−x^2$
$I_{rs}=\sqrt{\frac{1}{2a}\underset{−a}{\overset{+a}{∫}}{y}^{2}dx}$
$I_{rms}^2=\frac{1}{2a}\underset{−a}{\overset{+a}{∫}}\left(a^2−x^2\right)dx$
$=\frac{1}{2a}\underset{−a}{\overset{+a}{∫}}\left(a^2−x^2\right)dx=\frac{1}{2a}{âa^{2}x−\frac{x^3}{3}â}_{−a}^{+a}$
$=\frac{1}{2a}\left(a^3−\frac{a^3}{3}+a^3−\frac{a^3}{3}\right)=\frac{2a^2}{3}$
$I_{rms}=\sqrt{\frac{2a^2}{3}}=\sqrt{\frac{2}{3}}a$