In Fig. 32-38 a, switch S has been closed on A long enough...

In Fig. $32-38 a$, switch $\mathrm{S}$ has been closed on $A$ long enough to establish a steady current in the inductor of inductance $L_{1}=5.00 \mathrm{mH}$ and the resistor of resistance $R_{1}=$ $25 \Omega$. Similarly, in Fig. $32-38 b$, switch $\mathrm{S}$ has been closed on $A$ long enough to establish a steady current in the inductor of inductance $L_{2}=3.00 \mathrm{mH}$ and the resistor of resistance $R_{2}=30 \Omega$. The ratio $\Phi_{02} / \Phi_{01}$ of the magnetic flux through a turn in inductor 2 to that in inductor 1 is $1.5$. At time $t=0$, the two switches are closed on $B$. At what time $t$ is the flux through a turn in the two inductors equal?

Key Concepts

RL Circuit Transients

RL circuits are circuits that contain resistors and inductors and are characterized by their time-dependent behavior when a switch is opened or closed. In these circuits, the current does not change instantaneously; instead, it evolves gradually due to the energy stored in the magnetic field of the inductor, leading to transient responses that are typically governed by differential equations.

Time Constant

The time constant in an RL circuit, defined as ? = L/R, is a measure of the speed at which the currents and the energies stored in the circuit change. This constant gives the characteristic time over which the circuit’s transient response decays exponentially towards its steady state, and it is critical in predicting the behavior of the circuit at any given time after a switching event.

Magnetic Flux in an Inductor

Magnetic flux in an inductor is directly related to the current flowing through it, with the relationship being defined by ? = L × I, where L is the inductance and I is the instantaneous current. This concept is essential when analyzing how energy is stored in the magnetic field of an inductor and how it evolves during transient processes in the circuit.

Exponential Decay of Current

In an RL circuit after a switch action, the current decays (or grows) exponentially over time according to the function I(t) = I? e^(?t/?) or a similar expression for the relevant transient period. This exponential nature is crucial for determining the behavior of the circuit’s magnetic flux, since the current—and hence the flux—decays exponentially, allowing for analysis and calculations regarding the time at which certain conditions, such as equal flux values, are met.

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