The current output from of an a.c. source is given as I = 10 sin to t. The d.c equivalent of the current is

Analyze the question

The current is I = 10 sin ωt, so the peak current is I₀ = 10 A. The "d.c. equivalent" of an a.c. is the steady current that produces the same heating effect. This is the root mean square (r.m.s.) value.

Concept

For a sinusoidal a.c.:

I_rms = I₀ / √2 = 0.707 × I₀

This comes from the average of sin² over a full cycle, which is ½. So the mean power is ½ I₀²R, and the equivalent d.c. current satisfies I²R = ½ I₀²R, giving I = I₀/√2.

Solution

I_rms = 10 / √2 I_rms = 10 / 1.414 I_rms ≈ 7.07 A ≈ 7.1 A

Answer: Option B (7.1 A).

Why the other options are wrong

• A (5.0 A) is I₀/2, which is not the r.m.s. value.
• C (10.0 A) is the peak value itself.
• D (14.1 A) comes from multiplying by √2 instead of dividing.

Exam tip

"Equivalent d.c." means r.m.s. Just remember: r.m.s. = peak ÷ √2 ≈ 0.707 × peak. Dividing 10 by 1.414 gives about 7.1, so the answer must be smaller than the peak. This lets you eliminate C and D immediately.