🔄 Magnetic Flux

Part 1 of 7 — Counting Field Lines Through a Surface

In the 1830s, Michael Faraday discovered that changing magnetic fields can produce electric currents. To quantify this, we first need a way to measure "how much magnetic field passes through a surface." That quantity is magnetic flux.

Defining Magnetic Flux

Magnetic flux $\Phi_B$ measures the total magnetic field passing through a given area. Think of it as counting "how many field lines thread through a loop."

Formula

$\Phi_B = BA\cos\theta$

where:

• $B$ = magnetic field strength (T)
• $A$ = area of the surface (m²)
• $\theta$ = angle between $\vec{B}$ and the area normal $\hat{n}$ (the vector perpendicular to the surface)

SI Unit

$[\Phi_B] = \text{T} \cdot \text{m}^2 = \text{Wb (Weber)}$

Visualizing Flux

Imagine rain falling on a hoop:

• Hoop flat (face up): Maximum rain passes through → $\theta = 0°$, $\Phi = BA$
• Hoop tilted: Less rain passes through → $0° < \theta < 90°$, $\Phi = BA\cos\theta$
• Hoop vertical (on edge): No rain passes through → $\theta = 90°$, $\Phi = 0$

The same logic applies to magnetic field lines passing through a loop of wire.

Special Cases of Flux

Case 1: $\theta = 0°$ — Field perpendicular to surface (parallel to normal)

$\Phi_B = BA\cos 0° = BA \quad \text{(maximum flux)}$

The field lines pass straight through the loop.

Case 2: $\theta = 90°$ — Field parallel to surface (perpendicular to normal)

$\Phi_B = BA\cos 90° = 0 \quad \text{(zero flux)}$

The field lines skim along the surface without passing through.

Case 3: $\theta = 60°$ — Tilted surface

$\Phi_B = BA\cos 60° = \frac{1}{2}BA$

Only half the maximum flux threads through the loop.

Important Sign Convention

Flux can be positive or negative depending on which direction the field passes through the surface. If $\vec{B}$ points in the same direction as $\hat{n}$, the flux is positive. If opposite, it's negative. For a single loop, we usually choose $\hat{n}$ so that flux is positive.

Multiple Loops (Coil)

For a coil with $N$ turns, the total flux linkage is:

$\Phi_{\text{total}} = N\Phi_B = NBA\cos\theta$

Each turn contributes the same flux, so we multiply by $N$.

Magnetic Flux Concept Check 🧠

Ways to Change Magnetic Flux

Since $\Phi_B = BA\cos\theta$, the flux changes if any of the three factors change:

1. Change $B$ (magnetic field strength)

• Slide a magnet toward or away from a loop
• Increase/decrease current in a nearby electromagnet

2. Change $A$ (area of the loop)

• Stretch or compress a flexible loop
• Pull a loop out of the field region

3. Change $\theta$ (angle between $\vec{B}$ and $\hat{n}$)

• Rotate the loop in the field
• This is how generators work!

Why This Matters

Faraday discovered that changing flux induces an EMF (voltage) in the loop. The faster the flux changes, the larger the induced EMF. This is the foundation of electromagnetic induction — the topic of this entire unit!

Magnetic Flux Calculation Drill 📐

A rectangular loop has dimensions 20 cm × 30 cm and sits in a uniform magnetic field of $B = 0.5$ T.

1. Area of the loop in m²
2. Maximum possible flux through the loop (in Wb)
3. Flux when the loop is tilted so $\theta = 60°$ (in Wb)

Round all answers to 3 significant figures.

Exit Quiz — Magnetic Flux ✅

⚡ Faraday's Law of Induction

Part 2 of 7 — EMF from Changing Flux

Michael Faraday's greatest discovery: a changing magnetic flux through a loop induces an electromotive force (EMF). This single law is the basis of generators, transformers, and most of the electrical power grid.

Faraday's Law

Statement

The induced EMF in a loop is equal to the negative rate of change of magnetic flux through the loop:

$\varepsilon = -\frac{d\Phi_B}{dt}$

For a coil with $N$ turns:

$\varepsilon = -N\frac{d\Phi_B}{dt}$

Key Points

• The magnitude of the EMF depends on how fast the flux changes
• The negative sign is related to Lenz's Law (Part 3) — it tells us the direction of the induced EMF opposes the change
• If the flux is constant ($d\Phi_B/dt = 0$), there is no induced EMF
• The unit of EMF is the Volt (V)

Average EMF

For a finite change in flux over a time interval:

$|\varepsilon| = N\frac{|\Delta\Phi_B|}{\Delta t} = N\frac{|\Phi_f - \Phi_i|}{\Delta t}$

This is the form you'll use most often in AP Physics 2 calculations.

Three Ways to Induce an EMF

Since $\Phi_B = BA\cos\theta$, the flux can change by changing any factor:

1. Changing $B$ — Varying the Field

Push a magnet toward a coil: $B$ increases at the coil → $\Phi$ increases → EMF induced.

$|\varepsilon| = NA\cos\theta \cdot \frac{|\Delta B|}{\Delta t}$

Example: A solenoid's field increases from 0 to 0.5 T in 0.1 s through a 100-turn coil of area 0.02 m² ($\theta = 0°$):

$|\varepsilon| = (100)(0.02)(1)\frac{0.5}{0.1} = 10 \text{ V}$

2. Changing $A$ — Varying the Area

Pull a loop partially out of the field: the area inside the field decreases → $\Phi$ decreases → EMF induced.

$|\varepsilon| = NB\cos\theta \cdot \frac{|\Delta A|}{\Delta t}$

3. Changing $\theta$ — Rotating the Loop

Rotate the loop in the field: $\theta$ changes → $\cos\theta$ changes → $\Phi$ changes → EMF induced.

This is exactly how an AC generator works (covered in Part 5).

Faraday's Law Concept Check 🧠

Worked Example: Magnet Moving Into a Coil

A bar magnet is pushed into a 200-turn coil of radius 5 cm. The magnetic field at the coil increases uniformly from 0 T to 0.4 T in 0.25 s.

Step 1: Find the area

$A = \pi r^2 = \pi(0.05)^2 = 7.85 \times 10^{-3} \text{ m}^2$

Step 2: Find the change in flux (per turn)

$\Delta\Phi = \Delta B \cdot A \cdot \cos 0° = (0.4)(7.85 \times 10^{-3})(1) = 3.14 \times 10^{-3} \text{ Wb}$

Step 3: Apply Faraday's Law

$|\varepsilon| = N\frac{|\Delta\Phi|}{\Delta t} = 200 \times \frac{3.14 \times 10^{-3}}{0.25} = 200 \times 0.01256 = 2.51 \text{ V}$

Key Takeaway

Even modest changes in flux through many-turn coils can produce significant voltages!

Faraday's Law Calculation Drill 📐

A square coil has 80 turns, each with side length 10 cm. The coil sits in a uniform field ($\theta = 0°$) that decreases from 0.6 T to 0.2 T in 0.05 s.

1. Area of each turn (in m²)
2. Change in flux per turn $|\Delta\Phi|$ (in Wb)
3. Magnitude of the induced EMF (in V)

Round all answers to 3 significant figures.

Exit Quiz — Faraday's Law ✅

🔁 Lenz's Law

Part 3 of 7 — Nature Opposes the Change

Faraday's Law tells us the magnitude of the induced EMF. Lenz's Law tells us the direction. It embodies a profound principle: nature resists changes in magnetic flux.

Lenz's Law — Statement

The induced current flows in a direction that opposes the change in magnetic flux that produced it.

This is the physical meaning of the negative sign in Faraday's Law:

$\varepsilon = -N\frac{d\Phi_B}{dt}$

What "Opposes the Change" Means

• If flux is increasing → the induced current creates a magnetic field that opposes the external field (to try to prevent the increase)
• If flux is decreasing → the induced current creates a magnetic field in the same direction as the external field (to try to prevent the decrease)

The Key Insight

The induced current doesn't oppose the flux itself — it opposes the change in flux. If the flux is constant, there is no induced current at all.

Step-by-Step Method for Finding Induced Current Direction

Step 1: Determine the direction of the external magnetic field ($\vec{B}_{\text{ext}}$) through the loop.

Step 2: Determine whether the flux is increasing or decreasing.

• Is $B$ getting stronger/weaker?
• Is the loop moving into/out of the field?
• Is the loop area growing/shrinking?

Step 3: Find the direction of the induced magnetic field ($\vec{B}_{\text{ind}}$).

• Flux increasing → $\vec{B}_{\text{ind}}$ opposes $\vec{B}_{\text{ext}}$
• Flux decreasing → $\vec{B}_{\text{ind}}$ is in the same direction as $\vec{B}_{\text{ext}}$

Step 4: Use the right-hand rule to find the current direction.

• Curl the fingers of your right hand in the direction of $\vec{B}_{\text{ind}}$ through the loop
• Your curled fingers point in the direction of the induced current

Example: North pole of a magnet approaches a loop

1. $\vec{B}_{\text{ext}}$ points toward the loop (from the N pole)
2. Flux is increasing (magnet getting closer)
3. $\vec{B}_{\text{ind}}$ must oppose → points away from the magnet
4. The loop acts like a magnet with its N pole facing the approaching magnet — it repels the magnet!

Classic Lenz's Law Scenarios

Magnet Approaching a Loop

• North pole approaches → flux into loop increases → induced current creates field pointing back at magnet → loop's near face becomes North → repels the magnet

Magnet Retreating from a Loop

• North pole moves away → flux into loop decreases → induced current creates field pointing toward magnet → loop's near face becomes South → attracts the magnet

Both Cases: The Loop Opposes the Motion!

This is a consequence of energy conservation. If the induced current aided the motion, the magnet would accelerate, generating more current, generating more force — creating energy from nothing. That would violate conservation of energy!

Eddy Currents

When a solid conductor moves through a non-uniform magnetic field (or a changing field passes through a conductor), loops of current form within the bulk of the metal. These are eddy currents.

By Lenz's Law, eddy currents always create forces that oppose the relative motion — this is the principle behind:

• Magnetic braking (used in roller coasters and trains)
• Metal detectors
• Induction cooktops (eddy currents generate heat)
• Electromagnetic damping in galvanometers

Lenz's Law Concept Check 🧠

Lenz's Law Direction Drill 🧭

For each scenario, determine the direction of the induced current (as viewed from the specified side).

Exit Quiz — Lenz's Law ✅

🚂 Motional EMF

Part 4 of 7 — Moving Conductors in Magnetic Fields

When a conductor moves through a magnetic field, the free charges inside experience a magnetic force. This force drives a current — producing what we call motional EMF. It's Faraday's Law in action, derived from the Lorentz force.

The Sliding Rod Setup

Imagine a conducting rod of length $L$ sliding with velocity $v$ along two parallel rails connected by a resistor $R$. A uniform magnetic field $\vec{B}$ points into the page.

Deriving the EMF

As the rod moves to the right with speed $v$, the area of the circuit increases:

$\frac{dA}{dt} = L \cdot v$

The flux is increasing:

$\frac{d\Phi}{dt} = B \cdot \frac{dA}{dt} = BLv$

By Faraday's Law:

$|\varepsilon| = BLv$

This is the motional EMF for a rod moving perpendicular to both its own length and the magnetic field.

The Induced Current

$I = \frac{\varepsilon}{R} = \frac{BLv}{R}$

By Lenz's Law, the current flows counterclockwise (to oppose the increasing into-page flux).

Force on the Moving Rod

The current-carrying rod sits in a magnetic field, so it experiences a force:

$F = BIL = B \cdot \frac{BLv}{R} \cdot L = \frac{B^2L^2v}{R}$

Direction of the Force

By Lenz's Law (or the $\vec{F} = I\vec{L} \times \vec{B}$ force law), this force opposes the rod's motion — it acts to the left if the rod moves right.

Constant Velocity Requires an External Force

To keep the rod moving at constant velocity, you must apply an external force equal and opposite to the magnetic braking force:

$F_{\text{ext}} = \frac{B^2L^2v}{R}$

Power Analysis

$P_{\text{ext}} = F_{\text{ext}} \cdot v = \frac{B^2L^2v^2}{R}$

$P_{\text{dissipated}} = I^2R = \left(\frac{BLv}{R}\right)^2 R = \frac{B^2L^2v^2}{R}$

The power you put in equals the power dissipated as heat in the resistor. Energy is conserved! Mechanical energy → electrical energy → thermal energy.

Motional EMF Concept Check 🧠

Rail Problem — Complete Analysis

Problem Setup

A 0.5 m long rod slides at 4 m/s along frictionless rails connected by a 2 Ω resistor. The field is $B = 0.3$ T into the page.

Solution

EMF: $\varepsilon = BLv = (0.3)(0.5)(4) = 0.6 \text{ V}$

Current: $I = \frac{\varepsilon}{R} = \frac{0.6}{2} = 0.3 \text{ A}$

Magnetic braking force: $F = BIL = (0.3)(0.3)(0.5) = 0.045 \text{ N}$

Or equivalently: $F = \frac{B^2L^2v}{R} = \frac{(0.3)^2(0.5)^2(4)}{2} = \frac{0.09 \times 0.25 \times 4}{2} = 0.045 \text{ N}$

Power to maintain constant speed: $P = Fv = (0.045)(4) = 0.18 \text{ W}$

Power dissipated in resistor: $P = I^2R = (0.3)^2(2) = 0.18 \text{ W} \quad \checkmark$

Motional EMF Calculation Drill 📐

A conducting rod of length 0.8 m slides at 5 m/s along rails connected to a 4 Ω resistor in a uniform field $B = 0.5$ T (perpendicular to the rail plane).

1. Induced EMF (in V)
2. Current in the circuit (in A)
3. Force needed to maintain constant velocity (in N)

Round all answers to 3 significant figures.

Exit Quiz — Motional EMF ✅

🔌 Generators and Transformers

Part 5 of 7 — Turning Motion into Electricity (and Vice Versa)

The generator is arguably humanity's most important invention. By spinning a coil in a magnetic field, we convert mechanical energy into electrical energy. Transformers then allow us to transmit that power efficiently across vast distances.

The AC Generator

How It Works

A coil with $N$ turns and area $A$ rotates at angular frequency $\omega$ in a uniform field $\vec{B}$.

As the coil rotates, the angle $\theta = \omega t$, and the flux through the coil changes:

$\Phi_B = NBA\cos(\omega t)$

The Generator EMF

Applying Faraday's Law:

$\varepsilon = -\frac{d\Phi_B}{dt} = NBA\omega\sin(\omega t)$

The peak EMF is:

$\varepsilon_0 = NBA\omega$

So the output voltage oscillates sinusoidally:

$\varepsilon(t) = \varepsilon_0 \sin(\omega t)$

Key Features

• The output is alternating current (AC) — it reverses direction every half-cycle
• The frequency of the AC equals the rotation frequency: $f = \omega / (2\pi)$
• In the US, power plants produce AC at $f = 60$ Hz, so $\omega = 120\pi$ rad/s
• Peak EMF increases with $N$, $B$, $A$, and $\omega$

Generator Concept Check 🧠

Transformers

A transformer transfers AC electrical energy between two coils using electromagnetic induction. It consists of:

• Primary coil: $N_1$ turns, connected to the AC source
• Secondary coil: $N_2$ turns, connected to the load
• Iron core: channels the magnetic flux so nearly all flux through the primary also passes through the secondary

The Transformer Equation

Since both coils share the same changing flux:

$\frac{V_2}{V_1} = \frac{N_2}{N_1}$

Types of Transformers

Power Conservation

For an ideal transformer (no energy loss):

$P_1 = P_2 \implies V_1 I_1 = V_2 I_2$

Combining with the voltage equation:

$\frac{I_1}{I_2} = \frac{N_2}{N_1}$

If you step up the voltage, you step down the current — and vice versa. You cannot get more power out than you put in!

Why Transformers Matter

Power lines use high voltage ($\sim$500 kV) to reduce current, which reduces $I^2R$ resistive losses in the wires. Without step-up/step-down transformers, long-distance power transmission would be impractical.

Transformer Calculation Drill 📐

A step-up transformer has 200 turns on the primary and 5000 turns on the secondary. The primary is connected to a 120 V AC source supplying 10 A.

1. Secondary voltage $V_2$ (in V)
2. Secondary current $I_2$ (in A)
3. Power delivered to the load (in W)

Round all answers to 3 significant figures.

Power Transmission — Why High Voltage?

The Problem

A power plant generates 1 MW of power. The transmission lines have total resistance $R = 10\;\Omega$.

At Low Voltage (1000 V)

$I = \frac{P}{V} = \frac{10^6}{1000} = 1000 \text{ A}$

$P_{\text{lost}} = I^2 R = (1000)^2(10) = 10{,}000{,}000 \text{ W} = 10 \text{ MW!}$

That's 10× more than the power being transmitted! Totally impractical.

At High Voltage (500,000 V)

$I = \frac{P}{V} = \frac{10^6}{500{,}000} = 2 \text{ A}$

$P_{\text{lost}} = I^2 R = (2)^2(10) = 40 \text{ W}$

Only 0.004% lost! This is why we use high-voltage power lines.

The Full System

1. Generator produces AC at moderate voltage
2. Step-up transformer raises voltage to ~500 kV for transmission
3. Long-distance power lines carry small current
4. Step-down transformer reduces voltage to 120/240 V for homes

Exit Quiz — Generators & Transformers ✅

🧲 Inductance

Part 6 of 7 — Self-Induction and Energy Storage

A changing current in a coil produces a changing magnetic field, which produces a changing flux — through the same coil. By Faraday's Law, this induces an EMF that opposes the current change. This phenomenon is called self-inductance, and it gives coils a kind of electrical "inertia."

Self-Inductance

Definition

The self-inductance $L$ of a coil relates the flux through the coil to the current producing it:

$N\Phi_B = LI$

The SI unit of inductance is the Henry (H):

$1 \text{ H} = 1 \frac{\text{Wb}}{\text{A}} = 1 \frac{\text{V} \cdot \text{s}}{\text{A}}$

Induced EMF Due to Self-Inductance

Taking the time derivative of $N\Phi_B = LI$:

$\varepsilon = -L\frac{dI}{dt}$

This says: the faster the current changes, the larger the induced EMF opposing the change.

Key Properties

• $L$ depends only on the geometry of the coil (number of turns, area, length, core material) — not on the current
• The negative sign means the induced EMF always opposes the change in current (Lenz's Law)
• An inductor resists changes in current, just as a capacitor resists changes in voltage

Inductance of a Solenoid

For an ideal solenoid with $N$ turns, length $\ell$, cross-sectional area $A$, and core permeability $\mu$:

$L = \frac{\mu N^2 A}{\ell}$

More turns, larger area, shorter length → higher inductance.

Inductance Concept Check 🧠

Energy Stored in an Inductor

Building up current in an inductor requires work against the self-induced EMF. This work is stored as energy in the magnetic field:

$U = \frac{1}{2}LI^2$

Comparison with a Capacitor

Energy Density

The energy per unit volume stored in a magnetic field:

$u_B = \frac{B^2}{2\mu_0}$

This is the magnetic counterpart to the electric field energy density $u_E = \frac{1}{2}\varepsilon_0 E^2$.

RL Circuits

An RL circuit contains a resistor $R$ and inductor $L$ in series.

Charging (Switch Closed, Current Growing)

When you connect a battery of EMF $\varepsilon_0$ to an RL circuit:

$I(t) = \frac{\varepsilon_0}{R}\left(1 - e^{-t/\tau}\right)$

where the time constant is:

$\tau = \frac{L}{R}$

Key Behavior

• At $t = 0$: $I = 0$ (inductor blocks sudden current change)
• At $t = \tau$: $I = 0.632 \times \varepsilon_0/R$ (63.2% of max)
• At $t \to \infty$: $I = \varepsilon_0/R$ (inductor acts like a wire)

Discharging (Battery Removed, Current Decaying)

$I(t) = I_0 e^{-t/\tau}$

The current decays exponentially with the same time constant $\tau = L/R$.

Analogy to RC Circuits

Inductance & RL Circuit Drill 📐

An RL circuit has $L = 0.2$ H and $R = 10\;\Omega$, connected to a 20 V battery.

1. Time constant $\tau$ (in s)
2. Maximum (steady-state) current (in A)
3. Energy stored in the inductor at steady state (in J)

Round all answers to 3 significant figures.

Exit Quiz — Inductance ✅

🎯 Synthesis & AP Review

Part 7 of 7 — Putting It All Together

This final part combines Faraday's Law, Lenz's Law, motional EMF, generators, transformers, and inductance into comprehensive problems. We'll also highlight the most common AP mistakes and preview the types of free-response questions you'll encounter.

Master Equation Sheet — Electromagnetic Induction

The Big Picture

All of electromagnetic induction flows from one principle: a changing magnetic flux induces an EMF. Lenz's Law gives the direction. Everything else (motional EMF, generators, transformers, inductors) is a specific application of this idea.

Common AP Mistakes to Avoid ⚠️

Mistake 1: Confusing flux with field

• $\vec{B}$ is the field (a vector, in Tesla)
• $\Phi_B$ is the flux (a scalar, in Weber)
• A strong field doesn't mean large flux — it depends on area and angle too!

Mistake 2: Forgetting the angle in flux

• $\theta$ is between $\vec{B}$ and the area normal $\hat{n}$, NOT between $\vec{B}$ and the surface
• If the field is "perpendicular to the loop" → $\theta = 0°$ (field is parallel to $\hat{n}$)
• If the field is "parallel to the loop" → $\theta = 90°$

Mistake 3: Using Lenz's Law incorrectly

• The induced current opposes the change in flux, not the flux itself
• If flux is increasing, the induced field opposes the external field
• If flux is decreasing, the induced field reinforces the external field

Mistake 4: Confusing EMF with current

• Faraday's Law gives the EMF (voltage), not the current
• To find current, you need: $I = \varepsilon / R$
• An open-circuit loop has induced EMF but zero current

Mistake 5: Transformers and DC

• Transformers only work with AC (need changing flux)
• $V_2/V_1 = N_2/N_1$ applies to AC amplitudes or RMS values
• Power is conserved: stepping up voltage steps down current

Synthesis Quiz — Connecting the Concepts 🧠

AP Free-Response Preview 📝

Typical FRQ Structure

AP Physics 2 electromagnetic induction FRQs often combine multiple concepts in one problem:

Part (a): Calculate the magnetic flux at a given instant

Use $\Phi = BA\cos\theta$ and identify each quantity

Part (b): Find the induced EMF

Use $\varepsilon = -N\Delta\Phi/\Delta t$ — state Faraday's Law explicitly

Part (c): Determine the direction of induced current

Apply Lenz's Law — explain your reasoning step by step

Part (d): Calculate power or force

Use $P = \varepsilon^2/R$ or $F = BIL$

Scoring Tips

1. State the law you're using before applying it
2. Show your work — partial credit is common
3. Include units at every step
4. For Lenz's Law, explain the reasoning (flux increasing/decreasing → induced field direction → current direction)
5. Circle or box your final answer

Comprehensive Direction & Concept Drill 🧭

Test your understanding of the entire electromagnetic induction unit.

Final Mastery Quiz — Electromagnetic Induction 🏆