At time t=0 ms, a 45 V potential difference is suddenly...

At time $t=0 \mathrm{~ms}$, a $45 \mathrm{~V}$ potential difference is suddenly applied to the leads of a coil with inductance $L=50 \mathrm{mH}$ and resistance $R=180 \Omega$. At what rate is the current through the coil increasing at $t=1.2 \mathrm{~ms}$ ?

Key Concepts

Time Constant

The time constant in an RL circuit, denoted by tau (?) and calculated as L/R, is a measure of the speed at which the circuit responds to changes. It represents the time required for the current to reach about 63% of its final value, reflecting the rate of exponential growth during the transient phase.

Differential Equations in RL Circuits

The behavior of an RL circuit is governed by a first-order differential equation derived from Kirchhoff’s voltage law. This equation, which relates the applied voltage, the resistance, and the inductance, must be solved to determine the time-varying current and its rate of change during the transient phase.

Inductor Behavior

In an RL circuit, the inductor plays a crucial role by resisting rapid changes in the current flowing through it. This is due to its property of inductance, which is the ability of the coil to store energy in its magnetic field, causing the current to increase gradually rather than instantaneously when a voltage is applied.

RL Circuit

An RL circuit is a type of electrical circuit that includes a resistor and an inductor. When a voltage is applied suddenly, the circuit does not reach steady state immediately due to the inductor’s opposition to changes in current, resulting in a transient response that can be modeled by a differential equation.

Transient Response

The transient response of an RL circuit refers to its short-term behavior immediately after a sudden change in voltage. This response is characterized by a time-dependent exponential change of the current, as the circuit transitions from an initial state towards a new steady state.

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