Figure 30-30 gives the variation with time of the potential difference $\mathit{V}\mathit{R}$across a resistor in three circuits wired as shown in Fig. 30-16. The circuits contain the same resistance $\mathit{R}$and emf $\mathit{\epsilon }$but differ in the inductance L . Rank the circuits according to the value of L, greatest first.

1. Fig.30-30.
2. Fig.30-16.
3. All the circuits contain the same resistor R and emf $\epsilon$.
4. All the circuits contain different inductances L.

Using Eq.30-41 and 30-42, predict the value of inductance Lfrom potential difference V_R vs time t; that is from Fig.30-30. From the valueL , find theranks of the given circuit.

Formulae are as follows:

$T_L=\frac{L}{R}$

From Eq.30-41, the current is,

$i=\frac{\epsilon }{R}\left(1‐e^{-t/T_L}\right)..................................................................................\left(30-41\right)$

Where,the inductive time constant is given by,

$T_L=\frac{L}{R}..................................................................................................\left(30-42\right)$

From Eq.30-42, the higher value of inductance L causes the potential differenceV_R across the resistance R to take more time to reach its maximum and the lower value of inductance L causes the potential differenceV_R across the resistance R to take less time to reach its maximum.

From Fig.30-30, curve a gives the potential differenceV_R takes less time to reach its maximum as compared to that in curve b, and also, curve b gives the potential differenceV_R takes less time to reach its maximum as compared to that curve c. Therefore, the value of inductance L is more in curve c as compared with that in curve b and also the value of inductance L is more in curve b as compared with that in curve a.

Therefore, the ranks of the given circuit according to the value of L are c, b, and a.

Using Eq.30-41 and 30-42, and applying the properties of RL circuits, answer this question.