Introduction to LR Circuits
In our previous studies, we looked at how Resistors (R) control current and how Capacitors (C) store charge. Now, we are introducing the Inductor (L) into the mix. When you combine a resistor and an inductor in a circuit, you get an LR Circuit.
The "big idea" here is inertia. Just as a heavy object resists changes in its motion, an inductor resists changes in the electric current flowing through it. If you try to start a current, the inductor fights to keep it at zero. If you try to stop a current, the inductor fights to keep it moving. This chapter explores the math and physics behind that "fight."
1. The Setup: Powering Up (Current Growth)
Imagine a circuit with a battery (emf \(\mathcal{E}\)), a resistor (\(R\)), and an inductor (\(L\)) connected in series with a switch. When the switch is closed at \(t = 0\), the current doesn't jump to its maximum value immediately. Why? Because the changing current creates a changing magnetic flux, which induces a "back emf" in the inductor that opposes the battery.
Deriving the Expression using Calculus
We start with Kirchhoff’s Loop Rule. Walking around the loop, the sum of the potential differences must be zero:
\(\mathcal{E} - IR - L\frac{dI}{dt} = 0\)
To find the current \(I\) as a function of time \(t\), we rearrange this into a first-order differential equation:
\(L\frac{dI}{dt} = \mathcal{E} - IR\)
\(\frac{dI}{\mathcal{E} - IR} = \frac{1}{L} dt\)
Integrating both sides from time \(0\) to \(t\) (and current \(0\) to \(I\)):
\(\int_{0}^{I} \frac{dI'}{\mathcal{E} - I'R} = \int_{0}^{t} \frac{1}{L} dt'\)
After performing the integration and using some algebra (specifically involving natural logs and exponentials), we arrive at the Current Growth Equation:
\(I(t) = \frac{\mathcal{E}}{R}(1 - e^{-Rt/L})\)
Quick Review of Behavior:
- At \(t = 0\): \(I = \frac{\mathcal{E}}{R}(1 - 1) = 0\). The inductor acts like an open switch (infinite resistance).
- At \(t \to \infty\): \(I = \frac{\mathcal{E}}{R}(1 - 0) = \frac{\mathcal{E}}{R}\). The inductor acts like a plain wire (ideal inductor has zero resistance).
2. The Time Constant (\(\tau\))
How fast does the current reach its maximum? This depends on the Time Constant, denoted by the Greek letter tau (\(\tau\)).
\(\tau = \frac{L}{R}\)
What does \(\tau\) tell us?
- It is the time required for the current to reach approximately 63% of its maximum value (\(I_{max} = \mathcal{E}/R\)).
- The units of \(L/R\) are seconds (s). (Try a unit analysis: Henrys / Ohms = Seconds!)
- A large \(L\) means a "heavier" electromagnetic inertia, so it takes longer for current to change. A large \(R\) means the final current is smaller, so it reaches that state faster.
3. Powering Down (Current Decay)
What if the circuit has been running for a long time (so current is steady at \(I_0 = \mathcal{E}/R\)) and we suddenly bypass the battery? The inductor won't let the current vanish instantly. It will use its stored magnetic energy to keep the current flowing, though it will eventually "die out" due to the resistor dissipating energy as heat.
The Decay Equation
Applying the Loop Rule without a battery:
\(-IR - L\frac{dI}{dt} = 0\)
\(\frac{dI}{I} = -\frac{R}{L} dt\)
Integrating gives us the Current Decay Equation:
\(I(t) = I_0 e^{-Rt/L}\)
Key Takeaway: The current drops exponentially. After one time constant (\(t = \tau\)), the current has dropped to about 37% of its initial value.
4. Visualizing LR Circuits: Graphs
On the AP Exam, you are often asked to sketch or identify graphs of current or voltage vs. time.
- Current (Growth): Starts at zero and curves upward, leveling off at an asymptote of \(I = \mathcal{E}/R\).
- Voltage across the Inductor (\(V_L\)): Since \(V_L = L\frac{dI}{dt}\), the voltage is maximum the instant the switch is closed (it fights the hardest at the start) and decays to zero as the current becomes steady.
- Voltage across the Resistor (\(V_R\)): Since \(V_R = IR\), this graph looks exactly like the current graph (starts at 0, grows to \(\mathcal{E}\)).
Did you know?
The sparking you sometimes see when you pull a plug out of a wall is actually an LR circuit effect! The air becomes part of the circuit, and the inductor (like the windings in a motor) tries so hard to keep the current flowing that it ionizes the air to create a spark.
5. Summary and Common Pitfalls
Summary Table:
| Scenario | Equation | \(t=0\) behavior | \(t \to \infty\) behavior |
|---|---|---|---|
| Growth | \(I = I_{max}(1 - e^{-t/\tau})\) | \(I = 0\) | \(I = \mathcal{E}/R\) |
| Decay | \(I = I_0 e^{-t/\tau})\) | \(I = I_0\) | \(I = 0\) |
Common Mistakes to Avoid:
1. Mixing up \(\tau\): Don't confuse it with the RC time constant (\(\tau = RC\)). For inductors, it is \(\tau = L/R\).
2. The "Instant" Switch: Remember that current through an inductor cannot change instantaneously. If the current was 0.5 A the millisecond before you flipped a switch, it must be 0.5 A the millisecond after.
3. Ideal vs. Real: In AP problems, assume wires and inductors have zero resistance unless the problem states otherwise. If the inductor has internal resistance, treat it as an ideal inductor in series with an extra resistor.
Final Tip for the Exam: If you are asked to "justify" a behavior, always refer to Faraday's Law or Lenz's Law. The inductor induces an emf to oppose the change in magnetic flux caused by the changing current. That is the physics heart of every LR circuit question!