A flat coil of wire has an inductance of 40.0 mH and a...

A flat coil of wire has an inductance of $40.0 \mathrm{mH}$ and a resistance of $5.00 \Omega .$ It is connected to a $22.0-\mathrm{V}$ battery at the instant $t=0 .$ Consider the moment when the current is $3.00 \mathrm{~A}$. (a) At what rate is energy being delivered by the battery? (b) What is the power being delivered to the resistance of the coil? (c) At what rate is energy being stored in the magnetic field of the coil? (d) What is the relationship among these three power values? (e) Is the relationship described in part (d) true at other instants as well? (f) Explain the relationship at the moment immediately after $t=0$ and at a moment several seconds later.

A flat coil of wire has an inductance of $40.0 \mathrm{mH}$ and a resistance of $5.00 \Omega .$ It is connected to a $22.0-\mathrm{V}$ battery at the instant $t=0 .$ Consider the moment when the current is $3.00 \mathrm{~A}$. (a) At what rate is energy being delivered by the battery?
(b) What is the power being delivered to the resistance of the coil? (c) At what rate is energy being stored in the magnetic field of the coil? (d) What is the relationship among these three power values? (e) Is the relationship described in part (d) true at other instants as well? (f) Explain the relationship at the moment immediately after $t=0$ and at a moment several seconds later.

Key Concepts

Transient Response and Time-Dependent Behavior

The transient response of an RL circuit describes how the current, and therefore both the energy storage and power dissipation, changes immediately after a circuit is energized and over time until a steady state is reached. Immediately after the circuit is activated, the inductor limits the current rise, causing most of the battery power to be temporarily stored, whereas in the steady state, all battery power is dissipated by the resistance.

Power Conservation in Electrical Circuits

This principle states that the total power delivered by the battery in a circuit is partitioned between the power dissipated by the resistive elements and the rate of energy stored in the inductor’s magnetic field. Mathematically, this is expressed as the battery power (V*I) being equal to the sum of the resistor power (I^2R) and the rate of change of stored magnetic energy (d/dt(1/2 LI^2)).

RL Circuit Dynamics

This concept involves the behavior of electrical circuits containing both resistance (R) and inductance (L). In such circuits, when a voltage source is applied, the current does not immediately reach its maximum steady state but increases over time according to a differential equation. The inductance resists changes in current, leading to transient phenomena before the circuit reaches equilibrium.

Inductance and Energy Storage

Inductance is the property of a coil or circuit element that opposes changes in current flow by storing energy in its magnetic field. The energy stored in an inductor is given by (1/2) L I^2. This energy storage is dynamic and changes as the current varies over time, which is a critical aspect of transient behavior in circuits.

Resistive Dissipation

The resistance in a circuit dissipates electrical energy as heat according to Joule's law, with power dissipated given by I^2R. In any circuit, part of the battery's energy is applied to overcome this resistance, converting electrical energy irreversibly into thermal energy.

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