Question

A DC of $5$ produces the same heating effect as an A.C. of.

• A. $50$A rms current
• B. $5$A peak current
• C. $5$A rms current
• D. None of these

Answer 1

The correct option is C $5$A rms current

Let the rms value of current be $i_{rms}$.
So, instantaneous current can be written as $i=i_o\sin wt$
where $i_o=\sqrt{2}{i}_{rms}$
$⟹i=\sqrt{2}{i}_{rms}\sin wt$
Heat produced $H={\int }_o^T{i}^{2}Rdt=2i_{rms}^{2}R{\int }_o^T{\sin }^{2}wtdt$
For one time period, ${\int }_o^{T}si{n}^{2}wt=\frac{T}{2}$
Heat produced by DC of $5A$ in time $T$, $H_{DC}=5^{2}RT$
$\therefore$ $5^{2}RT=2i_{rms}^{2}R\frac{T}{2}$
$⟹i_{rms}=5A$
Thus $5A$ rms current produces same heat as that produced by DC of $5A$.