#AP Physics C: E&M - Unit 5: Electromagnetic Induction & Maxwell's Equations ⚡

#Overview 🧭

Hey there, future physicist! You've reached the final stretch of AP® Physics C: E&M! This unit is all about how electricity and magnetism intertwine, focusing on creating magnets with electricity, and exploring applications like motors, rail guns, and transformers. We'll also touch on Maxwell's Equations, the very foundation of all E&M concepts. This unit makes up a significant portion of the exam (14-20%), so let's make sure you're ready to ace it! 🚀

This unit accounts for 14-20% of the AP exam, making it a high-value topic. Make sure to review all the concepts thoroughly.

#5.1: Electromagnetic Induction

#Making Magnets from Electricity 🧙

Electromagnetic Induction is the process of generating a voltage using magnetic fields. If this voltage is in a complete circuit, it creates a current. Remember, in Unit 4, we saw how a current creates a magnetic field? Well, here we're doing the reverse! 🔄

Take a moment to explore this PhET simulation, especially the Pickup Coil tab. Notice what makes the bulb light up?

*Image created by the author using PhET*

The key is that the magnet must be moving! Just like a moving charge creates a magnetic field, a moving magnetic field induces a potential difference. 💡

#Magnetic Flux 🌐

Remember electric flux from Unit 2? Now, we're diving into magnetic flux to understand Faraday's Law. It’s calculated using a similar formula:

$$Φ_B = ∫ overrightarrow{B} ⋅ doverrightarrow{A}$$

*Image from electricalacademia.com*

Where:

• B is the magnetic field strength.
• dA is a tiny area chunk.
• θ is the angle between the magnetic field and area vectors.

The unit of magnetic flux is Tm², also known as a Weber (Wb). 📏

Magnetic flux can change in three ways:

*Image from physics.stackexchange.com*

• Changing the magnetic field strength (B).
• Changing the area (A) of the loop in the field.
• Changing the angle (θ) between the field and area.

#Faraday's Law & Lenz's Law 🔦

Faraday's Law tells us how a change in magnetic flux induces an electromotive force (EMF), which can drive a current: 💡

$$ε = -N \frac{dΦ_B}{dt}$$

• This formula applies to a single loop of wire.
• For multiple loops, we use N (number of loops), which is how transformers work.

Lenz's Law explains the negative sign in Faraday's Law. It gives the direction of the induced EMF (and thus the current). It states that the induced EMF always opposes the change in magnetic flux that caused it. 🧭

This is all due to the Law of Conservation of Energy. If the induced EMF aided the flux change, we'd get infinite energy! 🤯

Check out these cool videos for some visual examples: D!NG and Veritasium.

Let's look at some examples:

*Image from wikipedia*

• (a) Stationary magnet: No flux change, no current.
• (b) Magnet falling: Increasing flux. Induced B field opposes this, pointing upwards. Current is CCW (Right-Hand Rule).
• (c) Magnet moving away: Decreasing flux. Induced B field aids the original field, pointing downwards. Current is CW.

#Practical Applications ⚡

Transformers are used to step up or down voltages. Your phone charger uses a transformer to convert 120V (or 240V) from the wall to 5V for charging. 🔌

*Image from wikipedia*

The voltage change depends on the number of loops:

• More loops on the primary side = voltage step down.
• More loops on the secondary side = voltage step up.

Faraday's Law is also used in microphones and headphones. For more details, check out this infographic.

Motional EMF devices, like shake flashlights, generate EMF by moving an object in a magnetic field. No batteries needed! 🔦

*Image via amazon.com*

Here's how it works:

*Image from lumenlearning.com*

• (a) A metal bar moves right at velocity v, sweeping area A = lΔx.
• The increasing area increases the magnetic flux (ΦB = ∫ B ⋅ dA = BA).
• The induced field opposes this, pointing into the page.
• (b) Using RHR #2, the current flows CCW.

The induced EMF is:

$$ε = Blv$$

The induced current is:

$$I = \frac{Blv}{R}$$

#Maxwell's Equations 👨‍🔬

Maxwell's four equations are the foundation of all E&M! You've studied them all, even if you didn't know it. 😉 Here they are:

These equations describe how electric and magnetic fields interact and behave.

#Practice Problems ✅

*Images from collegeboard.org*

#Answers:

(a) The current is CCW. Two ways to see this:

• (i) Increasing flux into the page, so induced field is out of the page, making the current CCW.
• (ii) Positive charges in the rod are pushed upwards, so the current is CCW.

(b) Use Ohm's Law and Faraday's Law:

$$I = \frac{ε}{R} = \frac{Blv}{R}$$

(c)

$$F_B = ILB = \frac{B^2L^2v}{R}$$

(d) To keep the rod moving at constant speed, Fext = Fb. At t = 0, FB = B²L²v/R. As t increases, the graph should decrease.

*Image from collegeboard.org*

(e) The rod's speed will decrease. The magnetic force opposes the rod's velocity, causing negative acceleration.

Practice Question

Multiple Choice Questions

1. A conducting loop is placed in a uniform magnetic field. Which of the following will NOT induce a current in the loop? (A) Rotating the loop about an axis perpendicular to the field. (B) Changing the area of the loop. (C) Moving the loop parallel to the magnetic field. (D) Changing the strength of the magnetic field.

2. A transformer has 100 turns in its primary coil and 500 turns in its secondary coil. If the input voltage is 120 V, what is the output voltage? (A) 24 V (B) 600 V (C) 120 V (D) 240 V

Free Response Question

A rectangular loop of wire with width w and length L is placed in a uniform magnetic field B pointing into the page. The loop has a resistance R. The loop is pulled to the right at a constant speed v.

(a) Determine the direction of the induced current in the loop. (b) Derive an expression for the magnitude of the induced EMF in the loop. (c) Derive an expression for the magnitude of the induced current in the loop. (d) Derive an expression for the magnetic force acting on the loop. (e) What is the power required to keep the loop moving at a constant speed?

Answer Key:

Multiple Choice:

1. (C) Moving the loop parallel to the magnetic field
2. (B) 600 V

Free Response Question:

(a) The direction of the induced current is counterclockwise (CCW). [1 point] (b) The induced EMF is given by: ε = Blv [2 points] (c) The induced current is given by: I = ε/R = Blv/R [2 points] (d) The magnetic force acting on the loop is: F = ILB = (B²L²v)/R [2 points] (e) The power required is: P = Fv = (B²L²v²)/R [3 points]

#Final Exam Focus 🎯

Alright, you're in the home stretch! Here's what to focus on:

• Electromagnetic Induction: Understand how changing magnetic flux induces EMF. Practice using Faraday's and Lenz's Laws. 🧭
• Magnetic Flux: Know how to calculate it and how changes in B, A, and θ affect it. 🌐
• Motional EMF: Be ready to apply the formula ε = Blv in various scenarios. 🔦
• Transformers: Understand how they step up or down voltage based on the number of loops. ⚡
• Maxwell's Equations: Have a basic understanding of what each equation represents. 👨‍🔬

Last-Minute Tips:

• Right-Hand Rule: Master it! It's crucial for finding the direction of induced currents and forces. 🖐️
• Units: Always double-check your units! A small mistake can cost you points. 📏
• Practice: Go through the practice problems one more time. It’ll boost your confidence. ✅

You've got this! Go into that exam with confidence and show them what you've learned! 💪