An inductor that has an inductance of 15.0 H and a...

An inductor that has an inductance of $15.0 \mathrm{H}$ and a resistance of $30.0 \Omega$ is connected across a $100-\mathrm{V}$ battery. What is the rate of increase of the current
(a) at $t=0$ and
(b) at $t=1.50 \mathrm{~s}$ ?

Key Concepts

RL Circuit Analysis

RL circuits consist of a resistor and an inductor connected to a voltage source. They are fundamental in understanding transient behavior in electric circuits, as the inductor opposes changes in current. Analyzing such circuits involves setting up and solving differential equations that describe the time-dependent current response.

Time Constant

The time constant, defined as ? = L/R, is a crucial parameter in RL circuits. It indicates the time scale over which the current rises toward its final steady-state value. The time constant helps determine how quickly the circuit responds to changes, with larger values leading to slower responses and smaller values corresponding to more rapid changes.

Exponential Transient Response

The transient current in an RL circuit follows an exponential behavior as it transitions from zero to its steady-state value. This exponential response is characterized by the time constant and dictates the rate at which the current increases over time. It is key to calculating both instantaneous rates of change and overall current growth in the circuit.

Differential Equation Modeling

The behavior of current in an RL circuit is modeled by the differential equation V = L(dI/dt) + IR. This equation incorporates both the inductive element, which provides opposition to changes in current through L(dI/dt), and the resistive element, which dissipates energy through IR. Solving this equation is essential for determining the current’s rate of change at any point in time.

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