Question

Two resistors $R_1$ and $R_2$ may be connected either in series or in parallel across a battery of zero internal resistance. It is required that the joule heating for parallel combination be five times that for series combination. If $R_1=100\Omega$, find possible values of $R_2$.

• A. $38.2\Omega$
• B. $35.3\Omega$
• C. $48.2\Omega$
• D. $261.8\Omega$

Answer 1

Answer: (A), (D)

The correct options are
A $38.2\Omega$
D $261.8\Omega$

Equivalent resistance for parallel connection $R_p=\frac{100R_2}{100+R_2}$

Equivalent resistance for series connection $R_s=100+R_2$

Joule heat produced $H=\frac{V^{2}t}{R}$ $⟹H\propto \frac{1}{R}$

$\therefore$ $H_p=5H_s$ $⟹$ $\frac{1}{R_p}=\frac{5}{R_s}$

$\therefore$ $\frac{100+R_2}{100R_2}=\frac{5}{100+R_2}$

OR $R_2^2-300R_2+10000=0$

$⟹$ $R_2=38.2\Omega$ or $261.8\Omega$