A 25-mH inductor, an 8,0-Ωresistor, and a 6.0-V battery are connected in series. The switch is c

step 1 Given is a inductor which has inductance L equal to 25 millinery, given is a resistor which has

A 25-mH inductor, an 8,0-Ωresistor, and a 6.0-V battery are...

A $25-\mathrm{mH}$ inductor, an $8,0-\Omega$ resistor, and a $6.0-\mathrm{V}$ battery are connected in series. The switch is closed at $t=0$. Find the voltage drop across the resistor (a) at $t=0$ and (b) after one time constant has passed. Also, find the voltage drop across the inductor (c) at $t=0$ and $(\mathrm{d})$ after one time constant has elapsed.

Key Concepts

Series RL Circuit

A series RL circuit is one in which a resistor and an inductor are connected in series with a voltage source. This configuration exhibits transient behavior when the circuit is energized, as the inductor opposes changes in current. Analyzing such a circuit typically involves understanding how current and voltage changes evolve over time following a switching event.

Time Constant (?)

The time constant in an RL circuit is defined as ? = L/R, where L is the inductance and R is the resistance. The time constant is a measure of the speed at which current reaches its steady-state value after a sudden change, such as when a switch is closed. It characterizes the exponential approach of the current to its final value, with the circuit reaching approximately 63% of its final current after one time constant.

Inductor Behavior and Transient Response

Inductors resist sudden changes in current due to their property of self-inductance, which leads to an induced voltage that opposes the change in current. At the moment a circuit is energized (t = 0), the inductor initially behaves like an open circuit, maintaining zero current and causing the entire supply voltage to appear across it. Over time, as the current increases, the voltage across the inductor decreases exponentially, eventually becoming negligible in steady state.

Kirchhoff’s Voltage Law in Transient Circuits

Kirchhoff's Voltage Law states that the sum of the voltage drops around a closed loop equals the applied voltage. In a series RL circuit during transient response, this principle is used to relate the instantaneous voltage drops across the resistor and the inductor to the battery voltage. This framework is essential for setting up and solving the differential equations that describe the circuit's behavior over time.

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