At t=0, a battery is connected to a series arrangement of a...

At $t=0$, a battery is connected to a series arrangement of a resistor and an inductor. If the inductive time constant is $37.0$ $\mathrm{ms}$, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor's magnetic field?

Key Concepts

RL Circuit

An RL circuit is an electrical circuit consisting of a resistor and an inductor connected in series or parallel. It is used to study transient behaviors, as when a voltage is suddenly applied, the inductor resists changes in current and the circuit exhibits a delayed response. This is foundational for understanding how current evolves over time in circuits with inductance.

Time Constant

The time constant in an RL circuit, denoted by ? (tau), is defined as the ratio of the inductance L to the resistance R (? = L/R). It represents the characteristic time over which the circuit responds to changes, such as the rate at which current builds up or decays. The time constant is crucial in analyzing transient responses in electrical circuits.

Transient Response

The transient response of an RL circuit refers to the period immediately after a circuit parameter, such as voltage, is changed. During this time period, the current changes exponentially towards its steady state value. The mathematical description typically involves exponential growth or decay functions dependent on the time constant, which highlights the gradual adjustment of the system’s energy storage and dissipation.

Energy Storage in an Inductor

An inductor stores energy in its magnetic field, and the energy stored is given by the expression ½ L i², where i is the current through the inductor. As the current increases, the energy stored in the magnetic field grows accordingly. This concept is important for understanding how energy is temporarily held in a circuit before being fully transferred or dissipated.

Power Dissipation in a Resistor

A resistor dissipates electrical energy as heat, and the power dissipated in a resistor is given by the expression i²R, where i is the current and R is the resistance. Analyzing power dissipation is essential for ensuring proper thermal management in circuits and for understanding the distribution of energy between storage and loss mechanisms during transient events.

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