The current i in an electric circuit containing resistance R...

The current $i$ in an electric circuit containing resistance $R$ and inductance $L$ in series with a constant voltage source $E$ is given by the differential equation $E-L\left(\frac{\mathrm{d} i}{\mathrm{~d} t}\right)=R i$. Solve the equation and find $i$ in terms of time $t$ given that when $t=0, i=0$

Key Concepts

Transient Analysis in Electrical Circuits

Transient analysis deals with the behavior of electrical circuits as they respond to changes from an initial state to a steady state. In circuits with elements like resistors and inductors, the initial application of a voltage source results in a temporary period where the current changes over time, known as the transient response, before stabilizing.

First Order Linear Differential Equations

These are differential equations of the form dy/dt + P(t) y = Q(t), where the solutions can often be found using standard methods such as integrating factors. Understanding these equations is fundamental as they describe a wide range of systems in physics and engineering where the rate of change of a quantity depends linearly on the quantity itself.

Integrating Factor Method

This method is used to solve first order linear differential equations. It involves multiplying the entire equation by an integrating factor—typically an exponential function—that simplifies the left-hand side to the derivative of a product, thus enabling direct integration. This technique is a cornerstone in differential equations due to its wide applicability.

Time Constant

In the context of circuits, the time constant (often symbolized as ?) provides a measure of the speed at which the circuit responds to changes, such as reaching steady state. Specifically for LR circuits, it is defined as ? = L/R, where L is the inductance and R is the resistance. This parameter is crucial for predicting how quickly the transient effects decay.

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