The time t and electric current i for a certain circuit with...

The time $t$ and electric current $i$ for a certain circuit with a voltage $E,$ a resistance $R,$ and an inductance $L$ is given by $t=L \int \frac{d i}{E-i R}.$ If $t=0$ for $i=0,$ integrate and express $i$ as a function of $t.$

Key Concepts

RL Circuit Analysis

An RL circuit, which includes a resistor (R) and inductor (L) connected to a voltage source (E), is modeled by a differential equation derived from Kirchhoff’s voltage law. This differential equation characterizes how the current in the circuit changes over time as both the resistor and inductor influence the rate of current build?up.

Separation of Variables

Separation of variables is a technique used to solve differential equations by algebraically rearranging the terms so that each variable appears on a different side of the equation. In this context, the current-dependent term is isolated from the time-dependent term, enabling integration in order to find the functional dependence of current on time.

Integration of Rational Functions

When the differential equation for the RL circuit is separated, it often leads to an integral of a rational function, such as integrating din/(E ? iR). This integral typically results in a logarithmic function after performing a substitution, which is a common method in solving integrals that involve linear expressions in the denominator.

Exponential Response of RL Circuits

The integration process in analyzing RL circuits generally leads to a solution involving logarithms, which, when manipulated, results in an exponential function describing the time dependence of the current. This exponential behavior is characteristic of many first-order systems, capturing how the circuit asymptotically approaches steady state over a time constant defined by L and R.

Transcript

Please give Ace some feedback