Inductors & RL Circuits Detailed Cheat Sheet

A printable reference covering inductance, back emf, energy storage, RL time constants, current growth, and current decay for grades 11-12.

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Inductors are circuit elements that resist changes in current by creating a back emf. This cheat sheet covers how inductors behave, how energy is stored in magnetic fields, and how current changes in RL circuits. Students need these ideas to analyze switches, transients, and real circuits where current does not change instantly. The most important relationships connect induced voltage, inductance, current, and time. In an RL circuit, the time constant $\tau = \frac{L}{R}$ controls how quickly current rises or falls. Current growth follows $I(t) = I_{\max}\left(1 - e^{-t/\tau}\right)$ , while current decay follows $I(t) = I_0 e^{-t/\tau}$ .

Key Facts

The induced voltage across an inductor is $V_L = -L\frac{dI}{dt}$ , where the negative sign means the inductor opposes changes in current.
Inductance is measured in henries, with $1\ \text{H} = 1\ \frac{\text{V}\cdot\text{s}}{\text{A}}$ .
The energy stored in an inductor is $U = \frac{1}{2}LI^2$ .
For an RL circuit, the time constant is $\tau = \frac{L}{R}$ .
When current grows after a switch is closed, $I(t) = I_{\max}\left(1 - e^{-t/\tau}\right)$ and $I_{\max} = \frac{V}{R}$ .
When current decays after the source is removed, $I(t) = I_0 e^{-t/\tau}$ .
After one time constant during current growth, the current reaches about $63\%$ of its final value.
After one time constant during current decay, the current drops to about $37\%$ of its initial value.

Vocabulary

Inductor: A circuit component that stores energy in a magnetic field and opposes changes in current.
Inductance: The property of a component that measures how strongly it opposes changes in current, measured in henries.
Back emf: The induced voltage that acts against the change in current that produced it.
RL circuit: A circuit that contains resistance $R$ and inductance $L$ , causing current to change gradually over time.
Time constant: The time scale $\tau = \frac{L}{R}$ that determines how quickly current grows or decays in an RL circuit.
Transient response: The temporary changing behavior of voltage or current before a circuit reaches steady state.

Common Mistakes to Avoid

Treating current through an inductor as changing instantly is wrong because an inductor resists sudden current changes through back emf.
Using $\tau = RL$ instead of $\tau = \frac{L}{R}$ is wrong because the RL time constant increases with inductance and decreases with resistance.
Forgetting the negative sign in $V_L = -L\frac{dI}{dt}$ is wrong because the sign shows that the induced voltage opposes the change in current.
Using $I(t) = I_0 e^{-t/\tau}$ for current growth is wrong because that equation describes decay, not the rise toward $I_{\max}$ .
Assuming the inductor stores energy as electric potential energy is wrong because an inductor stores energy in a magnetic field.

Practice Questions

1 An RL circuit has $L = 0.50\ \text{H}$ and $R = 10\ \Omega$ . Find the time constant $\tau$ .
2 A $12\ \text{V}$ battery is connected to a series circuit with $R = 6.0\ \Omega$ and $L = 0.20\ \text{H}$ . Find $I_{\max}$ and $I(t)$ at $t = \tau$ .
3 An inductor with $L = 0.30\ \text{H}$ carries a current of $4.0\ \text{A}$ . How much energy is stored in its magnetic field?
4 Explain why a spark can occur when a switch is opened in a circuit containing an inductor.

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