A direct current of 5 A is superimposed on an alternating current \( I=10\sin\omega t \) (in A) flowing through a wire. The effective (rms) value of the resulting current is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 5\sqrt{3}\,\mathrm{A} \)
The rms (effective) current is the steady current that gives the same average heating \( I^{2}R \) in a resistor. Hence \( I^{2} \) equals the average of \( i^{2} \) over a cycle. Over one cycle the average of \( \sin\omega t \) is zero and the average of \( \sin^{2}\omega t \) is \( 1/2 \).
The current in the wire is \( i=5+10\sin\omega t \) (in A). The average of its square over a cycle is:
\( \begin{aligned} \langle i^{2}\rangle &= \langle(5+10\sin\omega t)^{2}\rangle \\ &= 25+100\,\langle\sin\omega t\rangle+100\,\langle\sin^{2}\omega t\rangle \\ &= 25+0+50 \\ &= 75\,\mathrm{A^{2}} \end{aligned} \)
The effective value of the current is:
\( \begin{aligned} I &= \sqrt{\langle i^{2}\rangle} \\ &= \sqrt{75}\,\mathrm{A} \\ &= 5\sqrt{3}\,\mathrm{A} \end{aligned} \)
NCERT (2026-27):
AC Voltage Applied to a Resistor – “To express ac power in the same form as dc power (P = I2R), a special value of current is defined and used. It is called, root mean square (rms) or effective current … and is denoted by Irms or I.”