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Combination of Resistors

Two heaters d...

Question

Two heaters designed for the same voltage V have different power ratings. When connected individually across a source of voltage V, they produce H amount of heat each in times $t_{1}$ and $t_{2}$ respectively. When used together across the same source, they produce H amount of heat in time $t$

If they are in series,

t = t_{1} + t_{2}

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If they are in series,

t = 2 (t_{1} + t_{2})

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If they are in parallel,

t = \frac{t_{1} t_{2}}{t_{1} + t_{2}}

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If they are in parallel,

t = \frac{t_{1} t_{2}}{2 (t_{1} + t_{2})}

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Solution

The correct options are
A If they are in series, $t = t_{1} + t_{2}$
C If they are in parallel, $t = \frac{t_{1} t_{2}}{t_{1} + t_{2}}$
Let resistances be $R_{1}$ and $R_{2}$
Then,

H $= \frac{V^{2} t_{1}}{R_{1}} = \frac{V^{2} t_{2}}{R_{2}}$ ...(1)
In series,

$R_{e q} = R_{1} + R_{2} ⟹ t = \frac{H \times R_{e q}}{V^{2}}$

replacing values of $R_{1} a n d R_{2}$ using (1)

we get $t = t_{1} + t_{2}$

in parallel, we get

$R_{e q} = \frac{R_{1} R_{2}}{R_{1} + R_{2}}$ ..(2)
again,

replacing values of $R_{1} a n d R_{2}$ using (1)

we get $t = \frac{t_{1} t_{2}}{t_{1} + t_{2}}$

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