This simple RL circuit is composed of a voltage source, an ohmic resistor, and an inductor. The voltage is expressed by the equation
is the voltage across the voltage source,
is the voltage across the resistor, and
is the voltage across the inductor.
According to Ohm's law, for any ohmic resistor,
is equal to
are the current and resistance through the resistor.
Current passing through an inductor causes magnetic flux, resulting in inductance. The inductance
is defined by
is the magnetic flux created by the current and
is the current through the inductor. Rearranging the terms gives
, which shows that any change in the current creates a change in the magnetic flux. According to Faraday's law of induction, a change in magnetic flux creates an induced voltage, represented by
. The negative sign comes from Lenz's law, which says that the induced voltage in an inductor always opposes the change in current.
Thus, we can replace
in the equation with
Solving the differential equation for the current
, we find
Special thanks to the University of Illinois NetMath Program and the Mathematics department at William Fremd High School.