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# What is the rms value of an alternating current which when passed through a resistor produces heat, which is three times that produced by direct current of $2A$ in the same resistor is(A) $6A$(B) $2A$(C) $3A$(D) $2\sqrt 3 A$

Last updated date: 14th Jul 2024
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The rms value of the current of an alternating current is responsible for producing heat. The formula for heat produced in resistor R, when I current is passed it from time it is $H = {I^2}Rt$.

Alternating current has fluctuating current amplitudes. If we have a DC supply with a value I, it will constantly provide the value I, but if we had a alternating current then we get a sinusoidal supply and sine has an average value of zero over a cycle therefore we use rms parameter is used.it is responsible for heating the device when alternating current passed through it.
Therefore, when passing alternating current through resistor R for time t heat will be equal to
${H_1} = I_{rms}^2Rt$
And we also know that for DC current, heat can be written as ${H_2} = {I^2}Rt = {2^2}Rt = 4Rt$
As it is given that the heat produced by the alternating current is three times to the heat produced by DC current. I.e. ${H_1} = 3{H_2}$
On substituting the values, we get
$\Rightarrow I_{rms}^2Rt = 3 \times 4Rt$
$\Rightarrow I_{rms}^2Rt = 12Rt$
$\Rightarrow I_{rms}^2 = \sqrt {12} = 2\sqrt 3 A$
Hence, the rms value of current is $2\sqrt 3 A$
Therefore, option (D) is correct.

There is a quantity called peak value of an alternating current defined as $I_0$.
This is the maximum value of current that it can achieve. It is related to rms as ${I_{rms}} = \dfrac{{{I_0}}}{{\sqrt 2 }}$.