Question

Determine the rms value of a semi-circular current wave which has a maximum value of a Ampere. The y axis denotes current in Ampere and x axis denotes time.

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

Answer 1

The correct option is C $\sqrt{\frac{2}{3}}a$

We know,
$I_{\text{rms}}=\sqrt{\frac{\int I^{2}dt}{\int dt}}$

Since, $x^2+y^2=a^2$
$\Rightarrow y^2=a^2-x^2$
Now,
$\int I^{2}dt\to {\int }_{-a}^a{y}^{2}dx$
$\Rightarrow {\int }_{-a}^a(a^2-x^2)dx={[a^{2}x-\frac{x^3}{3}]}_{-a}^a$
$\Rightarrow a^3-\frac{a^3}{3}+a^3-\frac{a^3}{3}=2a^3-\frac{2a^3}{3}$

Therfore,
$\int I^{2}dt=\frac{4a^3}{3}$
$\int dt\Rightarrow {\int }_{-a}^{a}dx=2a$
So,
$I_{\text{rms}}=\sqrt{\frac{2}{3}{a}^2}$
$I_{\text{rms}}=\sqrt{\frac{2}{3}}a$