Question

If the instantaneous current in a circuit is given by 2cos(t) ampere, the rms value of the current is :
(A) 2A
(B) $\sqrt 2 A$
(C) $2\sqrt 2 A$
(D) Zero

Answer 1

Hint: The rms current is defined as the square root of mean of square of the current. Mathematically it can be calculated by using the equation \[{i_{rms}} = \sqrt {\dfrac{{\int\limits_0^T {{i^2}dt} }}{T}} \] .

Complete step by step solution:
The equation describing rms current \[{i_{rms}} = \sqrt {\dfrac{{\int\limits_0^T {{i^2}dt} }}{T}} \], can also be written as \[{i_{rms}} = \sqrt {\left\langle {{i^2}} \right\rangle } \], where denotes the average of i2 for one complete cycle of current.
Since $i = 2\operatorname{Cos} t$ , we can say that average of i2 will be
\[\left\langle {{i^2}} \right\rangle = \left\langle {{2^2}{{\operatorname{Cos} }^2}t} \right\rangle \],
\[\left\langle {{i^2}} \right\rangle = 4\left\langle {{{\operatorname{Cos} }^2}t} \right\rangle \],
The average value of Cost for one complete cycle is zero, while the average value of Cos2t for one complete cycle can be remembered to be $\dfrac{1}{2}$ .
Hence, \[\left\langle {{i^2}} \right\rangle = 4 \times \dfrac{1}{2}\],
Solving, \[\left\langle {{i^2}} \right\rangle = 2\]
Since \[{i_{rms}} = \sqrt {\left\langle {{i^2}} \right\rangle } \],
Using the value \[\left\langle {{i^2}} \right\rangle = 2\],
We get, \[{i_{rms}} = \sqrt 2 \].

Therefore, the correct answer to the question is option : B

Additional information: The above described method can be used to calculate rms current for various types of variations of current or variation of voltage.
For simplicity, we can also remember that if current or voltage is defined according to the equation $y = {y_{\max }}\operatorname{Sin} (\omega t)$ , then the rms value of the current or voltage will be ${y_{rms}} = \dfrac{{{y_{\max }}}}{{\sqrt 2 }}$

Note: As written above, we use the direct result of calculation of rms value, we can see from the given current equation that the maximum value of current is 2 A. Hence the rms current can be directly written as
 ${i_{rms}} = \dfrac{{{i_{\max }}}}{{\sqrt 2 }}$,
${i_{rms}} = \dfrac{2}{{\sqrt 2 }}$,
${i_{rms}} = \sqrt 2 $.
This is a short cut method that is mostly made to remember. The significance of rms current is that when it is multiplied with resistance of the circuit, it can give us the power dissipated in the circuit in one complete cycle.