A coil of inductance 1 H and resistance 10 Ωis connected to a resistance-less battery of EMF 50

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A coil of inductance 1 H and resistance 10 Ωis connected to...

A coil of inductance $1 H$ and resistance $10 \Omega$ is connected to a resistance-less battery of EMF $50 \mathrm{~V}$ at time $t=0 .$ The ratio of rate at which magnetic energy is stored in the coil to the rate at which energy is supplied by the battery at $t=0.1 \mathrm{~s} .$ is $x \times 10^{-2}$. Find the value of $x$. (Given $\left.\frac{1}{e}=0.37\right)$

Key Concepts

Instantaneous Power in Circuit Elements

In circuits, instantaneous power is the rate at which energy is transferred or converted per unit time. For an RL circuit, the power supplied by the battery is divided into power dissipated as heat in the resistor and the rate of energy storage in the magnetic field of the inductor. Analyzing these power components helps in understanding energy conservation and losses during transient processes.

RL Circuit Analysis

RL circuits consist of resistors and inductors, and their behavior is governed by the interplay between resistive damping and inductive opposition to changes in current. Analyzing RL circuits involves understanding how current evolves over time when a voltage is applied, typically leading to exponential approaches to steady state.

Time Constant

The time constant in an RL circuit, defined as ? = L/R, characterizes the rate of change of current. It represents the time required for the current to reach approximately 63% of its final value. This parameter is essential for predicting both the transient response and dynamic energy storage in the circuit.

Magnetic Energy Storage

Inductors store energy in their magnetic fields, with the stored energy given by the formula U = (1/2)L I². This concept is fundamental in understanding how energy is accumulated during the transient behavior of an RL circuit as the current builds up over time.

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