At time t=0 ms, a 45.0 V potential difference is suddenly...

At time $t=0 \mathrm{~ms}$, a $45.0 \mathrm{~V}$ potential difference is suddenly applied to a coil with $L=50.0 \mathrm{mH}$ and $R=$ $180 \Omega$. At what rate is the current increasing at $t=1.20 \mathrm{~ms}$ ?

Key Concepts

Time Constant

The time constant in an RL circuit, defined as ? = L/R, is a measure of the rate at which the current reaches its final value after a voltage is applied. It characterizes the exponential rate of change in the current during the transient period of the circuit.

Rate of Change of Current

The instantaneous rate of change of current (di/dt) is a key aspect of transient analysis in RL circuits. It can be determined by differentiating the current equation or directly applying the differential form from Kirchhoff’s Law, providing insight into how rapidly the current increases or decreases at any given moment.

Differential Equations

The analysis of RL circuits relies on setting up and solving first-order differential equations derived from Kirchhoff's Voltage Law. These equations relate the applied voltage, the resistive drop, and the induced voltage in the inductor, thereby describing the evolution of current in the circuit.

RL Circuit Analysis

An RL circuit consists of a resistor and an inductor connected in series (or in another configuration), and its behavior is analyzed by considering how the inductor, which opposes changes in current, interacts with the resistor. The circuit’s voltage and current relationship is typically governed by Kirchhoff's Voltage Law, resulting in a differential equation that explains the transient and steady state response.

Exponential Response

When a step voltage is applied to an RL circuit, the current does not jump instantly to its final value. Instead, it rises exponentially according to the circuit’s time constant. This exponential behavior, which reflects the transient response, shows how the current gradually builds up over time.

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