A coil of some internal resistance 'r' behaves likes an inductance . When it is connected in series with a resistance $R_{1}$ ,the time constant is found to be $τ_{1}$ . When it is connected in series with a resistance $R_{2}$ the time constant is found to be $τ_{2}$ .Find $τ_{2}$ .

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Solution

R.E.F image
A coil has inductance L and resistance 'r'
Time constant for LR circuit, $T = \frac{L}{R} . \to (1)$
connected in series given : -
time constant $= r_{1}$
Equivalent resistance $= (r + R_{1})$ in seric
From (1), $r_{1} = \frac{L}{R_{1} + r}$
$R_{2}$ when connected in parallel.
Equivalent resistance $= \frac{R_{2} r}{R_{2} + r}$ in parallel
time constant $= r_{2}$
Thus, from (1),
$\Rightarrow r_{2} = \frac{L (R_{2} + r)}{R_{2} r} = L (\frac{1}{r} + \frac{1}{R_{2}})$

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A coil of some internal resistance 'r' behaves likes an inductance . When it is connected in series with a resistance R1,the time constant is found to be τ1. When it is connected in series with a resistance R2 the time constant is found to be τ2.Find τ2.

A coil of some internal resistance 'r' behaves likes an inductance . When it is connected in series with a resistance $R_{1}$ ,the time constant is found to be $τ_{1}$ . When it is connected in series with a resistance $R_{2}$ the time constant is found to be $τ_{2}$ .Find $τ_{2}$ .