📚

All Subjects

 > 

💡 

AP Physics C: E & M

 > 

⚛️

Unit 5

5.2 Inductance ➰

3 min read•november 3, 2020

caroline-koffke

Caroline Koffke

peter57616

Peter Apps


What is an Inductor? 🤔

An inductor is a coil of wire wrapped around a core that's used in electric circuits to smooth out DC current and resist sudden changes in current in the circuit. It does this by storing electrical energy in the form of magnetic fields.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-RHQms48TCTDp.jpg?alt=media&token=c1382ac0-b3ed-4e7d-8a13-b4b0c290624d

Image from wikipedia.org

We know from Faraday's Law that the amount of magnetic flux is proportional to the induced current. We'll define a new quantity L (inductance) as a constant of proportionality such that:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-08-29%20at%2010.44-YF3xnEEpg5BN.png?alt=media&token=0f220bfe-80bf-49d1-b7a6-19e83b9b67fc
Inductance is a measure of how much the inductor resists changes in its magnetic field. It can be altered by changing the number of loops in the coil, the gauge of the wire, and the material the core is made of. The SI unit for Inductance is the Henry (H).
Applying this new definition of magnetic flux to Faraday's Law, we can derive an equation for the induced EMF in terms of the current flowing through the inductor. Like before, this induced EMF opposes the change in flux.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-08-29%20at%2010.44-o0lzc6bJ3yIp.png?alt=media&token=cfb3d5a0-dcb9-4dff-968a-301840258253

Energy Stored in an Inductor 🔋

Let's assume you've constructed a fairly basic circuit as shown below:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-IBktypbFwVZ6.png?alt=media&token=2e77fe07-24df-49bb-8b89-81994b901f84
When we close the switch, the current will flow through the circuit, causing an increasing flux in the inductor. The inductor will then create a counter EMF to oppose the current in the circuit. Using Kirchoff's Voltage Law we can write the following equation:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-rb6uhaqoBmmr.png?alt=media&token=9c83f1af-911d-47e7-b521-d16a579e37e1
We have stored magnetic energy inside the inductor!

LR Circuit Behavior 🔍

Looking at the derivation from above, we chose to ignore the rest of the circuit to focus on the energy in the inductor. But what if we didn't...
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-IGs5kkwvWcPa.png?alt=media&token=bb259369-d71f-46b5-b47f-e4a9513511ff
This derivation is very similar to the capacitor equations we derived in Unit 3. In this case the time constant is equal to R / L.
We can also derive an expression for the voltage drop across the induction by again using Kirchoff's Voltage Law.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-fXlELbPIkvN7.png?alt=media&token=5de9ff63-9cdc-4f8a-9d36-96539ccdb32b
📊By graphing the equations we can determine the behavior of an inductor in the circuit.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-2JWDk1ssuOfa.gif?alt=media&token=4dbf3714-47c8-42b1-941c-ba2c0c6bc8b7

Image from electronics-tutorials.ws

When the switch is first closed, the inductor has a voltage equal to the battery. This initially prevents any current from flowing through the circuit. As t increases, the magnetic flux decreases, and more current is allowed to flow. Until we reach the steady-state value, the inductor is acting like a wire.

LC Circuit Behavior 🧲📸

The final type of circuit we need to look consists of nothing but a charged capacitor and an inductor.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-w58G5Kbob3D9.jpg?alt=media&token=141f2199-e8af-4682-977d-f175b9183c35

Image from phys.libretexts.org

In the beginning, all the charge is stored in the capacitor. It discharges, creating a current in the circuit (a). That current creates a changing magnetic flux in the inductor, which causes the inductor to store magnetic potential energy. Once the capacitor is discharged (b), the current drops and the inductor releases its energy to oppose the changing flux again.
This counter-current charges the capacitor (although with the opposite polarity) as shown in (c), then the cycle repeats. Image (e) shows the resulting oscillations in charge and current.
If we assume that there's no energy lost in the circuit (no resistance of any kind), then the total energy must remain constant.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-JYYMyVWqmxlG.png?alt=media&token=d7610098-9a5c-4944-a9cf-ecf9db859fc7
We can also use Faraday's Law to find the equations for current and charge as a function of time
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-JhhSRHQu31qt.png?alt=media&token=be12ae06-96c4-4bc1-9675-2c33cdb30889
To those who've taken AP C: Mech, this differential equation is the same form as the one you solved for simple harmonic motion. 
The solution is a function where the second derivative is the same as the function. These are trig functions!
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-RvJ1ZqMS4AwN.png?alt=media&token=e7b177f0-feed-49eb-b25e-6b91b5cdc71c

Practice Problem ✅

1)
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-uoDOls3fK13A.png?alt=media&token=625f4f18-3edd-4241-b586-7662c6e4318d

Image from collegeboard.org

Answers:

a) i) To those who've taken AP C: Mech, this differential equation is the same form as the one you solved for simple harmonic motion.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-08-29%20at%2010.47-nNsECs6WZoI6.png?alt=media&token=b8a6a212-f574-4d07-a8bf-c2f7f0301358
ii)An inductor acts like an open switch immediately after the switch is closed, so no current flows through the far right branch. Treat this like a series circuit.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-DO2cnQ3OEQjn.png?alt=media&token=9bd49807-026c-427b-94e0-0eb055a3b83a
iii) The voltage drop across the capacitor is 0 immediately after the switch is closed so we can ignore it. Treat this just like we did in part (i)
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-08-29%20at%2010.48-F7iTKGSz9EON.png?alt=media&token=77eff27c-80e1-4180-a511-9e12cd7829a3
b) The resistor will be a constant current since it's not time dependent. The capacitor will start with a very high current, then exponentially decrease. The inductor will begin allowing no current through then level off at some max current as
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-eRo9ZDO5FFT9.png?alt=media&token=82735c5d-0f4a-4cdb-9c1a-669818d6c282

Image from collegeboard.org

Was this guide helpful?

🔍 Are you ready for college apps?
Take this quiz and find out!
Start Quiz
FREE AP physics e m Survival Pack + Cram Chart PDF
Sign up now for instant access to 2 amazing downloads to help you get a 5
Browse Study Guides By Unit
🙏
Exam Reviews
✍️
Exam Skills- FRQ/MCQ
⚡️
Unit 1: Electrostatics
🔋
Unit 2: Conductors, Capacitors, Dielectrics
🔌
Unit 3: Electric Circuits
🧲
Unit 4: Magnetic Fields
Join us on Discord
Thousands of students are studying with us for the AP Physics C: E&M exam.
join now
Hours Logo
Studying with Hours = the ultimate focus mode
Start a free study session
📱 Stressed or struggling and need to talk to someone?
Talk to a trained counselor for free. It's 100% anonymous.
Text FIVEABLE to 741741 to get started.
© 2021 Fiveable, Inc.