Electro Magnetic Induction And Alternating Currents Question 167
Question: 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) $ (\sqrt{1/3})a $
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Answer:
Correct Answer: C
Solution:
-
The equation of a semi - circular wave is
$ x^{2}+y^{2}=a^{2} $
or $ y^{2}=a^{2}-x^{2} $
$ I _{rms}=\sqrt{\frac{1}{2a}\int _{-a}^{+a}{y^{2}dx}} $
$ I^{2} _{rms}=\frac{1}{2a}\int _{-a}^{+a}{(a^{2}-x^{2})dx} $
$ =\frac{1}{2a}\int _{-a}^{+a}{(a^{2}-x^{2}})dx=\frac{1}{2a}| a. ^{2}x-\frac{x^{3}}{3} | . _{-a}^{+a} $
$ =\frac{1}{2a}( a^{3}-\frac{a^{3}}{3}+a^{3}-\frac{a^{3}}{3} )=\frac{2a^{2}}{3} $
$ I _{rms}=\sqrt{\frac{2a^{2}}{3}}=\sqrt{\frac{2}{3}}a $
BETA
