RL Circuits
An RL circuit consists of an inductor and a resistor, in series or parallel with each other, with current driven by a voltage source.
Learning Objective

Describe currentvoltage relationship in the RL circuit and calculate energy that can be stored in an inductor
Key Points
 The energy stored in an inductor is
$E=\frac{1}{2}LI^{2}$ . It takes time to build up stored energy in a conductor and time to deplete it.  When a resistor and an inductor in series are connected to a voltage source, the timedependent current is given by
$I=I_{0}(1e^{\frac{t}{\tau}})$ . The final current after a long time is$I_0$ .  The characteristic time constant is given by
$\tau=\frac{L}{R}$ , where R is resistance and L is inductance. This represents the time necessary for the current in a circuit just closed to go from zero to$0.632\cdot I_0$ .  When the voltage source is disconnected from the inductor, the current will decay according to
$I=I_{0}e^{\frac{t}{\tau}}$ . In the first time interval τ the current falls by a factor of$\frac{1}{e}$ to$0.368\cdot I_0$ .
Terms

characteristic time constant
Denoted by
$\tau$ , in RL circuits it is given by$ \tau=\frac{L}{R}$ where R is resistance and L is inductance. When a switch is closed, it is the time it takes for the current to decay by a factor of 1/e. 
inductor
A device or circuit component that exhibits significant selfinductance; a device which stores energy in a magnetic field.
Full Text
RL Circuits
A resistorinductor circuit (RL circuit) consists of a resistor and an inductor (either in series or in parallel) driven by a voltage source.
Review
Recall that induction is the process in which an emf is induced by changing magnetic flux. Mutual inductance is the effect of Faraday's law of induction for one device upon another, while selfinductance is the the effect of Faraday's law of induction of a device on itself. An inductor is a device or circuit component that exhibits selfinductance.
Energy of an Inductor
We know from Lenz's law that inductors oppose changes in current. We can think of this situation in terms of energy. Energy is stored in a magnetic field. It takes time to build up energy, and it also takes time to deplete energy; hence, there is an opposition to rapid change. In an inductor, the magnetic field is directly proportional to current and to the inductance of the device. It can be shown that the energy stored in an inductor E_{ind} is given by:
Inductors in Circuits
We know that the current through an inductor L cannot be turned on or off instantaneously. The change in current changes the magnetic flux, inducing an emf opposing the change (Lenz's law). How long does the opposition last? Current will flow and can be turned off, but how long does it take? The following figure shows a switching circuit that can be used to examine current through an inductor as a function of time.
Current in an RL Circuit
(a) An RL circuit with a switch to turn current on and off. When in position 1, the battery, resistor, and inductor are in series and a current is established. In position 2, the battery is removed and the current eventually stops because of energy loss in the resistor. (b) A graph of current growth versus time when the switch is moved to position 1. (c) A graph of current decay when the switch is moved to position 2.
When the switch is first moved to position 1 (at t=0), the current is zero and it eventually rises to I_{0}=V/R, where R is the total resistance of the circuitand V is the battery's voltage. The opposition of the inductor L is greatest at the beginning, because the change in current is greatest at that time. The opposition it poses is in the form of an induced emf, which decreases to zero as the current approaches its final value. This is the hallmark of an exponential behavior, and it can be shown (with calculus) that
is the current in an RL circuit when switched on. (Note the similarity to the exponential behavior of the voltage on a charging capacitor.) The initial current is zero and approaches I_{0}=V/R with a characteristic time constant for an RL circuit, given by:
where
The characteristic time
When the switch in (a) is moved to position 2 and cuts the battery out of the circuit, the current drops because of energy dissipation by the resistor. However, this is also not instantaneous, since the inductor opposes the decrease in current by inducing an emf in the same direction as the battery that drove the current. Furthermore, there is a certain amount of energy, (1/2)LI_{0}^{2}, stored in the inductor, and it is dissipated at a finite rate. As the current approaches zero, the rate of decrease slows, since the energy dissipation rate is I^{2}R. Once again the behavior is exponential, and I is found to be
In (c), in the first period of time
Key Term Reference
 Component
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 Law
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 Lenz's law
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 battery
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 circuit
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 conductor
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 current
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 decay
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 energy
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 flux
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 induction
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 instantaneous
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 magnetic field
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 magnetic flux
 Appears in these related concepts: Inductance, Motional EMF, and Transformers
 mutual inductance
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 parallel
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 period
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 position
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 resistance
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 resistor
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 selfinductance
 series
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 voltage
 Appears in these related concepts: The Nernst Equation, Electric Potential Due to a Point Charge, and Principles of Electricity
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