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Question: The current in a circuit varies with time as \(I = 2\sqrt t \). The RMS value of the current for the...

The current in a circuit varies with time as I=2tI = 2\sqrt t . The RMS value of the current for the interval t=2st = 2s to t=4st = 4s is
A) 3A\sqrt 3 \,{\text{A}}
B) 23A2\sqrt 3 \,{\text{A}}
C) 32A\dfrac{{\sqrt 3 }}{2}\,{\text{A}}
D) (422)A\left( {4 - 2\sqrt 2 } \right){\text{A}}

Explanation

Solution

RMS value stands for the root-mean-square value of the varying current in a given time period. Square the value of current, normalize, and integrate it with respect to time over the specified period and take its square root.
Formula used: IRMS=1TI2dt{I_{RMS}} = \sqrt {\dfrac{1}{T}\int {{I^2}dt} } where II is the time-varying current in the circuit and TT is the interval of time

Complete step by step solution:
We’ve been given that the current in a circuit varies with time as I=2tI = 2\sqrt t . To find the root-mean-square value of the current in the given interval, we use
IRMS=1TI2dt\Rightarrow {I_{RMS}} = \sqrt {\dfrac{1}{T}\int {{I^2}dt} }
Since the time interval is t=2st = 2s to t=4st = 4s, T=42=2sT = 4 - 2 = 2s. So,
IRMS=12t=24(2t)2dt\Rightarrow {I_{RMS}} = \sqrt {\dfrac{1}{2}\int\limits_{t = 2}^4 {{{\left( {2\sqrt t } \right)}^2}dt} }
IRMS=2t=24tdt\Rightarrow {I_{RMS}} = \sqrt {2\int\limits_{t = 2}^4 {tdt} }
On carrying out the integration, we get
IRMS=2t22t=24\Rightarrow {I_{RMS}} = \sqrt {2\left. {\dfrac{{{t^2}}}{2}} \right|_{t = 2}^4}
Placing the values of t=2st = 2s and t=4st = 4s as the upper and lower limits of integration, we get
IRMS=(164)=12\Rightarrow {I_{RMS}} = \sqrt {(16 - 4)} = \sqrt {12}
So the value ofIRMS=12=23A{I_{RMS}} = \sqrt {12} = 2\sqrt 3 \,A which corresponds to option (B).

Additional Information:
For a sinusoidally current (AC circuit) in the circuit, the RMS value of current is IRMS=Io2{I_{RMS}} = \dfrac{{{I_o}}}{{\sqrt 2 }} where Io{I_o} is the maximum current in the circuit. The RMS value for an AC circuit holds more information about the circuit than the maximum current in the circuit. If the current in the circuit is constant with time, like in a DC circuit, the RMS value is equal to the constant value.

Note:
The root-mean-square current is a statistical quantity that is often used when dealing with varying currents in the circuit and is used in calculating the average power in the circuit. We can expect the RMS value of the current to lie between the maximum and the minimum value of the current in the circuit i.e. between 222\sqrt 2 and 242\sqrt 4 which can help us in removing option (A),(C),(D) as the possible answer.