Question

An alternating current having the peak value \( 10\sqrt{2}\,\text{A} \) is used to heat a metal wire. To produce the same heating effect, the constant current required is

• A. \(10\sqrt{2}\,\text{A} \)
• B. \(5\,\text{A} \)
• C. \(14\,\text{A} \)
• D. \(7\,\text{A} \)
• E. \(10\,\text{A} \)

Hint:

Heating effect depends on RMS value.

Answer 1

Answer: (E)

Step 1: Understanding Heating Effect and RMS Current:
The heating effect of a current is proportional to the square of the current (\(P = I^2R\)). Since an alternating current (AC) varies with time, its heating effect is not determined by its average value (which is zero) but by its effective value. This effective value is the Root Mean Square (RMS) current. The RMS value of an AC current is defined as the value of a constant direct current (DC) that would produce the same heating effect in the same resistor.
Step 2: Key Formula or Approach:
The question asks for the "constant current required" to produce the same heating effect, which is the definition of the RMS current (\(I_{rms}\)). The relationship between the RMS value and the peak value (\(I_{peak}\) or \(I_0\)) of a sinusoidal alternating current is:
\[ I_{rms} = \frac{I_{peak}}{\sqrt{2}} \] Step 3: Detailed Calculation:
We are given the peak value of the alternating current:
\[ I_{peak} = 10\sqrt{2} \, \text{A} \] Now, we can calculate the RMS value using the formula:
\[ I_{rms} = \frac{10\sqrt{2}}{\sqrt{2}} \] \[ I_{rms} = 10 \, \text{A} \] Step 4: Final Answer:
The constant current required to produce the same heating effect is equal to the RMS value of the AC current, which is 10 A.