Question

What is the approximate peak value of an alternating current producing four times the heat produced per second by a steady current of 2A in a resistor?
(a). 2.8A
(b). 4.0A
(c). 5.6A
(d). 8.0A

Answer 1

Hint: The heat produced per second by a current flowing through a resistance is equal to the product of the square of the current and the value of the resistance. The peak value of alternating current is $\sqrt 2 $ times the root mean square value of the alternating current.

Formula used:
The power dissipated by a current flowing through a resistor is given as follows:
$P = {I^2}R$
where P signifies the power dissipated, I represents the value of current flowing through the resistance having value R.
For alternating current, the peak value and the root mean square value of the current are related to each other by the following relation:
${I_{rms}} = \dfrac{{{I_0}}}{{\sqrt 2 }}$
where ${I_{rms}}$ represents the root mean square value of the alternating current while ${I_0}$ represents the peak value of the alternating current.

Complete answer:
Let we have a resistor of resistance R through which steady and alternating current are flowing as given.
First consider the steady current given to us. Its magnitude is given to be 2A, therefore, the power dissipated due to it in the resistor can be written as
${P_{steady}} = {\left( 2 \right)^2}R = 4R$
Now it is given to us that the power dissipated by an alternating current is four times that dissipated by the steady current, therefore, we can write
${P_{rms}} = 4{P_{steady}} = 16R{\text{ }}...{\text{(i)}}$
But we can write the power dissipated for alternating current as follows:
${P_{rms}} = I_{rms}^2R{\text{ }}...{\text{(ii)}}$
Comparing equations (i) and (ii), we get
$
  I_{rms}^2R = 16R \\
   \Rightarrow I_{rms}^2 = 16 \\
   \Rightarrow {I_{rms}} = 4A \\
$
Now we can easily calculate the peak value of alternating current from its rms value as follows:
${I_0} = \sqrt 2 {I_{rms}} = \sqrt 2 \times 4 = 5.656A \simeq 5.6A$
Hence, the correct answer is option d.

Note: 1. The peak value of the alternating current represents the maximum amount of current that can flow due to an alternating current.
2. When we talk about alternating current, we use its rms value for calculations not the peak value.