Electromagnetism 2

Lecture idea

This lecture is designed around concrete physical problems rather than abstract definitions alone. Each chapter corresponds directly to one problem from the list Lecture_Notes/Physics/problems_repo/problem_set_06_electromagnetism_ii.md.

The goal is to understand magnetism and electromagnetic induction through:

force acting on a moving charge,
motion in magnetic and crossed fields,
torque and energy of current loops,
generation of EMF by motion and rotation,
electrodynamic braking,
self-induction and energy stored in fields,
transformers and eddy currents.

Later, each chapter can be paired with a dedicated HTML animation placed in:

Lecture_Notes/Physics/html_anim/electromagnetism_2

What we want to understand

Before solving formulas, it helps to keep a few general questions in mind:

When does a magnetic field change motion, and when does it change energy?
How does motion in a magnetic field create voltage?
Why does induction oppose change?
How is mechanical energy converted into electrical energy and then into heat?
How do real devices such as generators, transformers, and magnetic brakes emerge from the same core principles?

1. Lorentz Force

This chapter should grow directly from Problem 1. It introduces the magnetic part of the Lorentz force and uses it to explain why charged particles bend in a magnetic field.

Main ideas

Magnetic force on a charge:

\[ \vec F = q \vec v \times \vec B \]

The direction of the force is perpendicular to both velocity and magnetic field.
Because the force is perpendicular to motion, the field bends the trajectory instead of increasing the speed.
In the simplest case of $\vec v \perp \vec B$, the motion becomes circular.

Questions for students

How do we determine the direction of the force from the cross product?
Why does the magnetic force not do work?
What sets the radius of the trajectory?
What changes when the field $B$ becomes stronger?

Planned animation

An animation should show a charged particle entering a uniform magnetic field with adjustable:

charge sign,
mass,
speed,
magnetic field strength.

The key visual point is the relation between $\vec v$, $\vec B$, and $\vec F$ and the emergence of circular motion.

First interactive app for this chapter

We now use a dedicated HTML animation for this first chapter:

Lecture_Notes/Physics/html_anim/electromagnetism_2/lorentz_force.html

The app is meant to be the visual foundation of the chapter. It shows the motion of a charged particle in the plane when the magnetic field is perpendicular to the screen.

What the student should see immediately:

the particle trajectory bends because of the Lorentz force,
the magnetic force is always perpendicular to the velocity,
changing the sign of the charge reverses the direction of curvature,
increasing $|B|$ decreases the radius of the orbit,
increasing the mass increases the radius,
the speed stays approximately constant, so the magnetic field changes direction of motion rather than kinetic energy.

What the app currently controls:

charge $q$,
mass $m$,
magnetic field component $B_z$,
initial velocity components $v_x(0)$ and $v_y(0)$,
animation speed,
zoom out of the visible trajectory window,
visibility of the velocity vector, force vector, and trajectory trail.

Implementation note for future work:

the numerical time evolution in Lecture_Notes/Physics/html_anim/electromagnetism_2/lorentz_force.html should use RK4 rather than a simple Euler-type step,
the reason is that circular trajectories in a uniform magnetic field should remain stable and visually close properly,
the zoom-out control is important because for weak fields or large mass the cyclotron radius becomes large and students should still be able to inspect the full orbit.

This app should be embedded directly here in the lecture:

html_anim/electromagnetism_2/lorentz_force.html

How we should extend this chapter later

When we return to this chapter, the next useful extensions are:

show the analytical formula for the cyclotron radius

\[ r = \frac{m v_\perp}{|q|B} \]

add a toggle for the case with a velocity component parallel to $\vec B$,
add a second panel with plots $x(t)$ and $y(t)$,
add a mode comparing positive and negative charges side by side,
add a short guided exercise using the exact data from Problem 1.

So this first app is not only an illustration. It is also the starting point for a small family of electromagnetism animations that can later support exercises and lecture discussion.

2. Velocity Selector in Crossed Fields

This chapter corresponds to Problem 2 and is a natural continuation of the Lorentz force. Here the electric and magnetic forces compete with each other.

Main ideas

Electric force:

\[ \vec F_E = q \vec E \]

Magnetic force:

\[ \vec F_B = q \vec v \times \vec B \]

Straight motion is possible when both forces cancel.
This gives the selected velocity:

\[ v_d = \frac{E}{B} \]

Questions for students

Why do only particles with one specific speed pass undeflected?
What happens to slower particles?
What happens to faster particles?
Does the kinetic energy change in the ideal steady regime?

Planned animation

An animation should show particles entering crossed fields and splitting according to their velocity. This can later become one of the most intuitive simulations in the lecture.

Interactive app for this chapter

We use a dedicated HTML animation for the velocity selector:

Lecture_Notes/Physics/html_anim/electromagnetism_2/velocity_selector.html

The purpose of this app is to show the central idea of the selector as clearly as possible:

one particle beam is slower than the selected speed,
one beam has exactly the selected speed,
one beam is faster than the selected speed.

This side-by-side comparison is pedagogically useful because students can immediately see that:

the electric field and magnetic field act in opposite vertical directions,
only one precise value of velocity leads to rectilinear motion,
slower particles bend toward the direction of the electric force,
faster particles bend toward the direction of the magnetic force.

The app currently lets us control:

electric field strength $E$,
magnetic field strength $B$,
animation speed,
zoom out of the visible region,
the speed factor of the slower beam,
the speed factor of the faster beam.

During the lecture, the main storyline should be:

write the two forces

\[ F_E = qE, \qquad F_B = qvB \]

explain that they point in opposite directions for the chosen geometry,
impose cancellation of forces,
obtain the selector condition

\[ v_d = \frac{E}{B} \]

use the animation to show why only that beam passes straight.

This app should be embedded directly here:

html_anim/electromagnetism_2/velocity_selector.html

How we should extend this chapter later

When we return to this chapter, the most useful next extensions are:

add a switch for positive versus negative charge,
add a mode with a continuous beam rather than only three representative particles,
add numerical readout of the separate electric and magnetic forces for each beam,
add a detector screen on the right side,
add a compact exercise mode using the data from Problem 2.

3. Magnetic Moment of a Current Loop

This chapter is based on Problem 3 and shifts attention from a single charge to an extended current-carrying object.

Main ideas

A current loop behaves like a magnetic dipole.
Its magnetic moment is:

\[ \vec \mu = N I S \, \hat n \]

In an external field the loop experiences torque:

\[ \vec M = \vec \mu \times \vec B \]

The associated potential energy is:

\[ U = - \vec \mu \cdot \vec B \]

Questions for students

Why does the loop tend to rotate?
For which angle is the torque maximal?
Which orientations are stable and unstable?
How is this related to compass needles, electric motors, and magnetic dipoles in matter?

Planned animation

An animation should show a loop in a uniform field with visible vectors $\vec \mu$, $\vec B$, and $\vec M$, plus an angle slider.

Interactive app for this chapter

We now use a dedicated HTML animation for the magnetic moment of a current loop:

Lecture_Notes/Physics/html_anim/electromagnetism_2/magnetic_moment_loop.html

The purpose of this app is to make the geometric content of the formulas visible:

the magnetic moment vector $\vec \mu$ is attached to the loop,
the external field $\vec B$ is fixed,
the torque depends on the angle between $\vec \mu$ and $\vec B$,
the loop tends to rotate toward the stable aligned configuration,
the anti-aligned configuration is unstable,
the torque is strongest near $90^\circ$.

What the student should be able to test immediately:

increasing $N$, $I$, or $S$ increases the magnetic moment,
increasing $B$ increases the torque,
the torque vanishes for parallel and anti-parallel configurations,
the stable position corresponds to minimum potential energy,
the unstable position corresponds to maximum potential energy.

The app currently lets us control:

number of turns $N$,
current $I$,
area $S$,
magnetic field strength $B$,
angle $\theta$ between $\vec \mu$ and $\vec B$,
animation speed.

This app should be embedded directly here:

html_anim/electromagnetism_2/magnetic_moment_loop.html

4. Rotating Loop and the Origin of AC Generation

This chapter follows Problem 4 and should be one of the central induction chapters. It connects changing flux with generated EMF.

Main ideas

Magnetic flux through the loop:

\[ \Phi = \vec B \cdot \vec S \]

For a rotating loop, the angle changes with time, so the flux changes periodically.
Faraday’s law gives the induced EMF:

\[ \mathcal{E}(t) = - \frac{d\Phi}{dt} \]

This is the foundation of electric generators.

Questions for students

Why does rotation in a constant magnetic field generate alternating voltage?
Why does the EMF amplitude grow with angular speed $\omega$?
What is changing physically: the field, the area, or the angle?

Planned animation

An animation should show a rotating loop, the changing projected area, the flux $\Phi(t)$, and the resulting sinusoidal EMF.

Interactive app for this chapter

We now use a dedicated HTML animation for the rotating loop and AC generation:

Lecture_Notes/Physics/html_anim/electromagnetism_2/rotating_loop_ac_generator.html

The app is designed to connect the geometry of rotation directly to the formulas for flux and induced EMF.

What the student should see immediately:

the loop rotates in a constant magnetic field,
the angle between the area vector and the field changes continuously,
the projected area perpendicular to the field oscillates,
the magnetic flux $\Phi(t)$ follows a cosine law,
the induced EMF $\mathcal{E}(t)$ follows a sine law and changes sign periodically,
increasing the angular speed $\omega$ increases the EMF amplitude.

What the app currently controls:

number of turns $N$,
loop area $S$,
magnetic field strength $B$,
angular speed $\omega$,
initial phase $\theta_0$,
animation speed,
camera yaw and pitch for the 3D scene.

This app should be embedded directly here:

html_anim/electromagnetism_2/rotating_loop_ac_generator.html

5. Motional EMF in a Moving Rod

This chapter is tied to Problem 5. It is a simple but powerful example showing that induction can arise directly from motion.

Main ideas

Charges inside a moving conductor feel the magnetic Lorentz force.
This charge separation creates a potential difference between the rod ends.
For perpendicular motion:

\[ \mathcal{E} = B L v \]

More generally, the effect depends on the component of velocity perpendicular to the magnetic field.

Questions for students

Why do charges pile up at opposite ends of the rod?
How does the effect depend on rod length?
What changes when motion is oblique rather than perpendicular?
Where does the energy associated with the induced voltage come from?

Planned animation

An animation should show a rod moving through a field, with charge separation building up in real time and the voltage changing as speed or angle changes.

Interactive app for this chapter

We now use a dedicated HTML animation for motional EMF in a moving rod:

Lecture_Notes/Physics/html_anim/electromagnetism_2/motional_emf_moving_rod.html

The purpose of this app is to make the source of the induced voltage visible:

the rod moves through a uniform magnetic field,
charges inside the rod feel the magnetic Lorentz force,
positive and negative charges separate toward opposite rod ends,
the polarity changes with the force direction,
the induced voltage scales with $B$, $L$, $v$, and the effective perpendicular geometry.

What the student should be able to test immediately:

doubling the rod length doubles the emf,
increasing the field increases the emf,
increasing the speed increases the emf,
reducing the angle weakens the effect,
for the perpendicular case the app reproduces

\[ \mathcal{E} = B L v \]

The app currently lets us control:

rod length $L$,
speed $v$,
motion angle $\alpha$,
magnetic field strength $B$,
animation speed,
camera yaw and pitch.

This app should be embedded directly here:

html_anim/electromagnetism_2/motional_emf_moving_rod.html

6. Sliding Rod on Rails: Electromagnetic Braking

This chapter corresponds to Problem 6 and is a major bridge between induction and dynamics.

Main ideas

Motion of the rod creates motional EMF.
The closed circuit produces current:

\[ I(v) = \frac{\mathcal{E}(v)}{R} \]

That current interacts with the magnetic field and produces a force opposing motion.
The system behaves like motion with damping proportional to velocity.
At long times, terminal velocity appears.

Questions for students

Why does the induced force oppose motion?
Why is the braking stronger for larger velocity?
How does the system settle to terminal speed?
How does the power balance connect gravity, electric current, and Joule heating?

Planned animation

An animation should show the rod sliding down rails, current appearing, braking force growing, and velocity approaching a terminal value.

Interactive app for this chapter

We now use a dedicated HTML animation for the sliding rod on rails:

Lecture_Notes/Physics/html_anim/electromagnetism_2/sliding_rod_braking.html

The goal of this app is to connect the full chain of ideas in one place:

motion along the rails creates motional EMF,
the closed circuit produces current,
the current generates a magnetic force opposite to the motion,
the braking force grows with speed,
the rod approaches a terminal velocity instead of accelerating without limit,
in the long-time regime the gravitational power input is converted into Joule heat.

What the student should see immediately:

the rod first accelerates because gravity wins,
as the speed increases, the induced current becomes larger,
the magnetic braking force becomes stronger and stronger,
the net acceleration decreases,
the velocity curve levels off near the terminal value.

The app currently lets us control:

mass $m$,
rail spacing $L$,
magnetic field strength $B$,
total resistance $R$,
incline angle $\alpha$,
animation speed.

This app should be embedded directly here:

html_anim/electromagnetism_2/sliding_rod_braking.html

7. Loop Entering a Magnetic Field Region

This chapter follows Problem 7 and is perfect for explaining Lenz’s law in a very visual way.

Main ideas

The loop experiences induction only while the magnetic flux is changing.
Entering the field changes the area of the loop that is immersed in the field.
While the loop is fully inside a uniform field, the flux becomes constant and the induced current disappears.

Questions for students

Why does current appear only during the entering or leaving phase?
Why is there a braking force while the flux changes?
Why does the current vanish when the entire loop is fully inside a uniform field?
How does this example differ from the moving rod?

Planned animation

An animation should show a rectangular loop being pulled into a finite field region, with simultaneous plots of overlap area, flux, EMF, current, and force.

Interactive app for this chapter

We now use a dedicated HTML animation for the loop entering a magnetic field region:

Lecture_Notes/Physics/html_anim/electromagnetism_2/loop_entering_field.html

The app is designed to make Lenz’s law visible in a step-by-step way:

the loop starts outside the field region,
during entry the overlap area grows,
the magnetic flux increases,
the induced EMF and current appear only in that changing-flux phase,
the magnetic force opposes the pulling motion,
once the whole loop is inside the uniform field, the current vanishes.

What the student should be able to read directly from the app:

the overlap area controls the flux,
in the entering phase the flux grows linearly in time,
this gives a constant induced EMF for constant pulling speed,
the braking force exists only while the overlap changes,
the required mechanical power matches the Joule heating.

The app currently lets us control:

loop width $a$,
loop height $b$,
magnetic field strength $B$,
resistance $R$,
pulling speed $v$,
animation speed.

This app should be embedded directly here:

html_anim/electromagnetism_2/loop_entering_field.html

8. Self-Induction and RL Decay

This chapter comes from Problem 8 and should emphasize that a circuit can oppose changes in its own current.

Main ideas

A changing current creates a changing magnetic field.
That changing magnetic field induces an EMF in the same circuit.
This is self-induction:

\[ \mathcal{E}_L = - L \frac{dI}{dt} \]

In an RL circuit, current does not jump instantly.
During decay:

\[ I(t) = I_0 e^{-t/\tau}, \qquad \tau = \frac{L}{R} \]

Questions for students

Why does the current not disappear immediately after disconnecting the source?
Why is energy stored in the magnetic field?
How is that energy later dissipated as heat?
Why can overvoltage appear when a circuit is suddenly opened?

Planned animation

An animation should show current build-up and decay in an RL circuit together with magnetic energy and the induced voltage across the coil.

Interactive app for this chapter

We now use a dedicated HTML animation for RL self-induction and decay:

Lecture_Notes/Physics/html_anim/electromagnetism_2/rl_decay_self_induction.html

The goal of this app is to show both phases of the problem in one place:

while the source is connected, the current builds up gradually,
the coil opposes the growth of current through self-induced EMF,
after disconnection, the source is removed but the current does not vanish instantly,
the coil reverses its voltage and keeps the current flowing,
the magnetic energy stored in the field decays into Joule heat.

What the student should be able to read directly from the app:

the steady current is $I_0 = U/R$,
the time scale is $\tau = L/R$,
charging and decay are both exponential processes,
the sign of the coil voltage changes between the two phases,
the stored energy is $W = \frac12 L I^2$ and decreases during decay.

The app currently lets us control:

inductance $L$,
resistance $R$,
source voltage $U$,
animation speed.

This app should be embedded directly here:

html_anim/electromagnetism_2/rl_decay_self_induction.html

9. Ideal and Real Transformer

This chapter is based on Problem 9 and should collect the induction ideas into one practical device.

Main ideas

Alternating current in the primary coil creates changing magnetic flux in the core.
That changing flux induces voltage in the secondary coil.
In the ideal picture:

\[ \frac{U_2}{U_1} = \frac{N_2}{N_1} \]

Real transformers have losses:
- resistive heating,
- eddy currents,
- hysteresis,
- leakage flux.

Questions for students

Where does the secondary voltage actually come from?
Why does a transformer require changing flux?
What makes a real transformer different from an ideal one?
Why are laminated cores used?

Planned animation

An animation should show the primary AC current, flux in the core, and the induced secondary voltage, with options to compare ideal and lossy behavior.

10. Eddy Currents and Contactless Braking

This final chapter follows Problem 10 and closes the lecture with a vivid engineering application.

Main ideas

Changing magnetic flux in a bulk conductor induces circulating currents.
These eddy currents generate magnetic effects that oppose the motion or the flux change.
The result can be contactless braking.

Questions for students

Why do eddy currents appear in a solid conductor even without a wire loop?
Which materials enhance the effect?
How can the braking force be increased?
What are the advantages and disadvantages of eddy-current brakes?

Planned animation

An animation should show a metal plate moving through a magnetic field, the induced current loops, and the resulting braking effect.

Final Summary

The full lecture can be read as one coherent story:

A magnetic field bends the motion of charges.
Electric and magnetic forces can balance to select velocity.
Currents in loops behave like magnetic dipoles.
Changing flux induces EMF.
Motion in a field can generate voltage directly.
Induced currents create forces that oppose motion.
Electromagnetic systems convert mechanical energy into electrical energy and heat.
The same principles explain generators, transformers, and magnetic brakes.

Suggested development order for animations

If we want the lecture to become highly visual as quickly as possible, a good order is:

Lorentz force
Velocity selector
Rotating loop
Moving rod
Sliding rod on rails
Loop entering a field region
RL self-induction
Transformer
Eddy currents
Magnetic moment of a loop

--- title: Electromagnetism 2 format: html: theme: flatly grid: body-width: 1000px margin-width: 250px toc: true toc-depth: 3 highlight-style: tango code-line-numbers: true code-fold: true code-summary: "Show the code" code-tools: true code-block-bg: "rgba(42, 174, 42, 0.02)" code-block-border-left: "#2aae2a" code-language-label: true css: styles.css math: mathjax self-contained: true other-links: - text: Main page href: https://dchorazkiewicz.github.io/Mathematics_Physics_Lectures --- ## Lecture idea This lecture is designed around concrete physical problems rather than abstract definitions alone. Each chapter corresponds directly to one problem from the list `Lecture_Notes/Physics/problems_repo/problem_set_06_electromagnetism_ii.md`. The goal is to understand magnetism and electromagnetic induction through: - force acting on a moving charge, - motion in magnetic and crossed fields, - torque and energy of current loops, - generation of EMF by motion and rotation, - electrodynamic braking, - self-induction and energy stored in fields, - transformers and eddy currents. Later, each chapter can be paired with a dedicated HTML animation placed in: `Lecture_Notes/Physics/html_anim/electromagnetism_2` ## What we want to understand Before solving formulas, it helps to keep a few general questions in mind: - When does a magnetic field change motion, and when does it change energy? - How does motion in a magnetic field create voltage? - Why does induction oppose change? - How is mechanical energy converted into electrical energy and then into heat? - How do real devices such as generators, transformers, and magnetic brakes emerge from the same core principles? ## 1. Lorentz Force This chapter should grow directly from Problem 1. It introduces the magnetic part of the Lorentz force and uses it to explain why charged particles bend in a magnetic field. ### Main ideas - Magnetic force on a charge: $$ \vec F = q \vec v \times \vec B $$ - The direction of the force is perpendicular to both velocity and magnetic field. - Because the force is perpendicular to motion, the field bends the trajectory instead of increasing the speed. - In the simplest case of $\vec v \perp \vec B$, the motion becomes circular. ### Questions for students - How do we determine the direction of the force from the cross product? - Why does the magnetic force not do work? - What sets the radius of the trajectory? - What changes when the field $B$ becomes stronger? ### Planned animation An animation should show a charged particle entering a uniform magnetic field with adjustable: - charge sign, - mass, - speed, - magnetic field strength. The key visual point is the relation between $\vec v$, $\vec B$, and $\vec F$ and the emergence of circular motion. ### First interactive app for this chapter We now use a dedicated HTML animation for this first chapter: `Lecture_Notes/Physics/html_anim/electromagnetism_2/lorentz_force.html` The app is meant to be the visual foundation of the chapter. It shows the motion of a charged particle in the plane when the magnetic field is perpendicular to the screen. What the student should see immediately: - the particle trajectory bends because of the Lorentz force, - the magnetic force is always perpendicular to the velocity, - changing the sign of the charge reverses the direction of curvature, - increasing $|B|$ decreases the radius of the orbit, - increasing the mass increases the radius, - the speed stays approximately constant, so the magnetic field changes direction of motion rather than kinetic energy. What the app currently controls: - charge $q$, - mass $m$, - magnetic field component $B_z$, - initial velocity components $v_x(0)$ and $v_y(0)$, - animation speed, - zoom out of the visible trajectory window, - visibility of the velocity vector, force vector, and trajectory trail. Implementation note for future work: - the numerical time evolution in `Lecture_Notes/Physics/html_anim/electromagnetism_2/lorentz_force.html` should use RK4 rather than a simple Euler-type step, - the reason is that circular trajectories in a uniform magnetic field should remain stable and visually close properly, - the zoom-out control is important because for weak fields or large mass the cyclotron radius becomes large and students should still be able to inspect the full orbit. This app should be embedded directly here in the lecture: <iframe src="html_anim/electromagnetism_2/lorentz_force.html" width="100%" height="980px" frameborder="0"></iframe> [html_anim/electromagnetism_2/lorentz_force.html](html_anim/electromagnetism_2/lorentz_force.html) ### How we should extend this chapter later When we return to this chapter, the next useful extensions are: 1. show the analytical formula for the cyclotron radius $$ r = \frac{m v_\perp}{|q|B} $$ 2. add a toggle for the case with a velocity component parallel to $\vec B$, 3. add a second panel with plots $x(t)$ and $y(t)$, 4. add a mode comparing positive and negative charges side by side, 5. add a short guided exercise using the exact data from Problem 1. So this first app is not only an illustration. It is also the starting point for a small family of electromagnetism animations that can later support exercises and lecture discussion. ## 2. Velocity Selector in Crossed Fields This chapter corresponds to Problem 2 and is a natural continuation of the Lorentz force. Here the electric and magnetic forces compete with each other. ### Main ideas - Electric force: $$ \vec F_E = q \vec E $$ - Magnetic force: $$ \vec F_B = q \vec v \times \vec B $$ - Straight motion is possible when both forces cancel. - This gives the selected velocity: $$ v_d = \frac{E}{B} $$ ### Questions for students - Why do only particles with one specific speed pass undeflected? - What happens to slower particles? - What happens to faster particles? - Does the kinetic energy change in the ideal steady regime? ### Planned animation An animation should show particles entering crossed fields and splitting according to their velocity. This can later become one of the most intuitive simulations in the lecture. ### Interactive app for this chapter We use a dedicated HTML animation for the velocity selector: `Lecture_Notes/Physics/html_anim/electromagnetism_2/velocity_selector.html` The purpose of this app is to show the central idea of the selector as clearly as possible: - one particle beam is slower than the selected speed, - one beam has exactly the selected speed, - one beam is faster than the selected speed. This side-by-side comparison is pedagogically useful because students can immediately see that: - the electric field and magnetic field act in opposite vertical directions, - only one precise value of velocity leads to rectilinear motion, - slower particles bend toward the direction of the electric force, - faster particles bend toward the direction of the magnetic force. The app currently lets us control: - electric field strength $E$, - magnetic field strength $B$, - animation speed, - zoom out of the visible region, - the speed factor of the slower beam, - the speed factor of the faster beam. During the lecture, the main storyline should be: 1. write the two forces $$ F_E = qE, \qquad F_B = qvB $$ 2. explain that they point in opposite directions for the chosen geometry, 3. impose cancellation of forces, 4. obtain the selector condition $$ v_d = \frac{E}{B} $$ 5. use the animation to show why only that beam passes straight. This app should be embedded directly here: <iframe src="html_anim/electromagnetism_2/velocity_selector.html" width="100%" height="960px" frameborder="0"></iframe> [html_anim/electromagnetism_2/velocity_selector.html](html_anim/electromagnetism_2/velocity_selector.html) ### How we should extend this chapter later When we return to this chapter, the most useful next extensions are: 1. add a switch for positive versus negative charge, 2. add a mode with a continuous beam rather than only three representative particles, 3. add numerical readout of the separate electric and magnetic forces for each beam, 4. add a detector screen on the right side, 5. add a compact exercise mode using the data from Problem 2. ## 3. Magnetic Moment of a Current Loop This chapter is based on Problem 3 and shifts attention from a single charge to an extended current-carrying object. ### Main ideas - A current loop behaves like a magnetic dipole. - Its magnetic moment is: $$ \vec \mu = N I S \, \hat n $$ - In an external field the loop experiences torque: $$ \vec M = \vec \mu \times \vec B $$ - The associated potential energy is: $$ U = - \vec \mu \cdot \vec B $$ ### Questions for students - Why does the loop tend to rotate? - For which angle is the torque maximal? - Which orientations are stable and unstable? - How is this related to compass needles, electric motors, and magnetic dipoles in matter? ### Planned animation An animation should show a loop in a uniform field with visible vectors $\vec \mu$, $\vec B$, and $\vec M$, plus an angle slider. ### Interactive app for this chapter We now use a dedicated HTML animation for the magnetic moment of a current loop: `Lecture_Notes/Physics/html_anim/electromagnetism_2/magnetic_moment_loop.html` The purpose of this app is to make the geometric content of the formulas visible: - the magnetic moment vector $\vec \mu$ is attached to the loop, - the external field $\vec B$ is fixed, - the torque depends on the angle between $\vec \mu$ and $\vec B$, - the loop tends to rotate toward the stable aligned configuration, - the anti-aligned configuration is unstable, - the torque is strongest near $90^\circ$. What the student should be able to test immediately: - increasing $N$, $I$, or $S$ increases the magnetic moment, - increasing $B$ increases the torque, - the torque vanishes for parallel and anti-parallel configurations, - the stable position corresponds to minimum potential energy, - the unstable position corresponds to maximum potential energy. The app currently lets us control: - number of turns $N$, - current $I$, - area $S$, - magnetic field strength $B$, - angle $\theta$ between $\vec \mu$ and $\vec B$, - animation speed. This app should be embedded directly here: <iframe src="html_anim/electromagnetism_2/magnetic_moment_loop.html" width="100%" height="940px" frameborder="0"></iframe> [html_anim/electromagnetism_2/magnetic_moment_loop.html](html_anim/electromagnetism_2/magnetic_moment_loop.html) ## 4. Rotating Loop and the Origin of AC Generation This chapter follows Problem 4 and should be one of the central induction chapters. It connects changing flux with generated EMF. ### Main ideas - Magnetic flux through the loop: $$ \Phi = \vec B \cdot \vec S $$ - For a rotating loop, the angle changes with time, so the flux changes periodically. - Faraday's law gives the induced EMF: $$ \mathcal{E}(t) = - \frac{d\Phi}{dt} $$ - This is the foundation of electric generators. ### Questions for students - Why does rotation in a constant magnetic field generate alternating voltage? - Why does the EMF amplitude grow with angular speed $\omega$? - What is changing physically: the field, the area, or the angle? ### Planned animation An animation should show a rotating loop, the changing projected area, the flux $\Phi(t)$, and the resulting sinusoidal EMF. ### Interactive app for this chapter We now use a dedicated HTML animation for the rotating loop and AC generation: `Lecture_Notes/Physics/html_anim/electromagnetism_2/rotating_loop_ac_generator.html` The app is designed to connect the geometry of rotation directly to the formulas for flux and induced EMF. What the student should see immediately: - the loop rotates in a constant magnetic field, - the angle between the area vector and the field changes continuously, - the projected area perpendicular to the field oscillates, - the magnetic flux $\Phi(t)$ follows a cosine law, - the induced EMF $\mathcal{E}(t)$ follows a sine law and changes sign periodically, - increasing the angular speed $\omega$ increases the EMF amplitude. What the app currently controls: - number of turns $N$, - loop area $S$, - magnetic field strength $B$, - angular speed $\omega$, - initial phase $\theta_0$, - animation speed, - camera yaw and pitch for the 3D scene. This app should be embedded directly here: <iframe src="html_anim/electromagnetism_2/rotating_loop_ac_generator.html" width="100%" height="1180px" frameborder="0"></iframe> [html_anim/electromagnetism_2/rotating_loop_ac_generator.html](html_anim/electromagnetism_2/rotating_loop_ac_generator.html) ## 5. Motional EMF in a Moving Rod This chapter is tied to Problem 5. It is a simple but powerful example showing that induction can arise directly from motion. ### Main ideas - Charges inside a moving conductor feel the magnetic Lorentz force. - This charge separation creates a potential difference between the rod ends. - For perpendicular motion: $$ \mathcal{E} = B L v $$ - More generally, the effect depends on the component of velocity perpendicular to the magnetic field. ### Questions for students - Why do charges pile up at opposite ends of the rod? - How does the effect depend on rod length? - What changes when motion is oblique rather than perpendicular? - Where does the energy associated with the induced voltage come from? ### Planned animation An animation should show a rod moving through a field, with charge separation building up in real time and the voltage changing as speed or angle changes. ### Interactive app for this chapter We now use a dedicated HTML animation for motional EMF in a moving rod: `Lecture_Notes/Physics/html_anim/electromagnetism_2/motional_emf_moving_rod.html` The purpose of this app is to make the source of the induced voltage visible: - the rod moves through a uniform magnetic field, - charges inside the rod feel the magnetic Lorentz force, - positive and negative charges separate toward opposite rod ends, - the polarity changes with the force direction, - the induced voltage scales with $B$, $L$, $v$, and the effective perpendicular geometry. What the student should be able to test immediately: - doubling the rod length doubles the emf, - increasing the field increases the emf, - increasing the speed increases the emf, - reducing the angle weakens the effect, - for the perpendicular case the app reproduces $$ \mathcal{E} = B L v $$ The app currently lets us control: - rod length $L$, - speed $v$, - motion angle $\alpha$, - magnetic field strength $B$, - animation speed, - camera yaw and pitch. This app should be embedded directly here: <iframe src="html_anim/electromagnetism_2/motional_emf_moving_rod.html" width="100%" height="980px" frameborder="0"></iframe> [html_anim/electromagnetism_2/motional_emf_moving_rod.html](html_anim/electromagnetism_2/motional_emf_moving_rod.html) ## 6. Sliding Rod on Rails: Electromagnetic Braking This chapter corresponds to Problem 6 and is a major bridge between induction and dynamics. ### Main ideas - Motion of the rod creates motional EMF. - The closed circuit produces current: $$ I(v) = \frac{\mathcal{E}(v)}{R} $$ - That current interacts with the magnetic field and produces a force opposing motion. - The system behaves like motion with damping proportional to velocity. - At long times, terminal velocity appears. ### Questions for students - Why does the induced force oppose motion? - Why is the braking stronger for larger velocity? - How does the system settle to terminal speed? - How does the power balance connect gravity, electric current, and Joule heating? ### Planned animation An animation should show the rod sliding down rails, current appearing, braking force growing, and velocity approaching a terminal value. ### Interactive app for this chapter We now use a dedicated HTML animation for the sliding rod on rails: `Lecture_Notes/Physics/html_anim/electromagnetism_2/sliding_rod_braking.html` The goal of this app is to connect the full chain of ideas in one place: - motion along the rails creates motional EMF, - the closed circuit produces current, - the current generates a magnetic force opposite to the motion, - the braking force grows with speed, - the rod approaches a terminal velocity instead of accelerating without limit, - in the long-time regime the gravitational power input is converted into Joule heat. What the student should see immediately: - the rod first accelerates because gravity wins, - as the speed increases, the induced current becomes larger, - the magnetic braking force becomes stronger and stronger, - the net acceleration decreases, - the velocity curve levels off near the terminal value. The app currently lets us control: - mass $m$, - rail spacing $L$, - magnetic field strength $B$, - total resistance $R$, - incline angle $\alpha$, - animation speed. This app should be embedded directly here: <iframe src="html_anim/electromagnetism_2/sliding_rod_braking.html" width="100%" height="1060px" frameborder="0"></iframe> [html_anim/electromagnetism_2/sliding_rod_braking.html](html_anim/electromagnetism_2/sliding_rod_braking.html) ## 7. Loop Entering a Magnetic Field Region This chapter follows Problem 7 and is perfect for explaining Lenz's law in a very visual way. ### Main ideas - The loop experiences induction only while the magnetic flux is changing. - Entering the field changes the area of the loop that is immersed in the field. - While the loop is fully inside a uniform field, the flux becomes constant and the induced current disappears. ### Questions for students - Why does current appear only during the entering or leaving phase? - Why is there a braking force while the flux changes? - Why does the current vanish when the entire loop is fully inside a uniform field? - How does this example differ from the moving rod? ### Planned animation An animation should show a rectangular loop being pulled into a finite field region, with simultaneous plots of overlap area, flux, EMF, current, and force. ### Interactive app for this chapter We now use a dedicated HTML animation for the loop entering a magnetic field region: `Lecture_Notes/Physics/html_anim/electromagnetism_2/loop_entering_field.html` The app is designed to make Lenz's law visible in a step-by-step way: - the loop starts outside the field region, - during entry the overlap area grows, - the magnetic flux increases, - the induced EMF and current appear only in that changing-flux phase, - the magnetic force opposes the pulling motion, - once the whole loop is inside the uniform field, the current vanishes. What the student should be able to read directly from the app: - the overlap area controls the flux, - in the entering phase the flux grows linearly in time, - this gives a constant induced EMF for constant pulling speed, - the braking force exists only while the overlap changes, - the required mechanical power matches the Joule heating. The app currently lets us control: - loop width $a$, - loop height $b$, - magnetic field strength $B$, - resistance $R$, - pulling speed $v$, - animation speed. This app should be embedded directly here: <iframe src="html_anim/electromagnetism_2/loop_entering_field.html" width="100%" height="1260px" frameborder="0"></iframe> [html_anim/electromagnetism_2/loop_entering_field.html](html_anim/electromagnetism_2/loop_entering_field.html) ## 8. Self-Induction and RL Decay This chapter comes from Problem 8 and should emphasize that a circuit can oppose changes in its own current. ### Main ideas - A changing current creates a changing magnetic field. - That changing magnetic field induces an EMF in the same circuit. - This is self-induction: $$ \mathcal{E}_L = - L \frac{dI}{dt} $$ - In an RL circuit, current does not jump instantly. - During decay: $$ I(t) = I_0 e^{-t/\tau}, \qquad \tau = \frac{L}{R} $$ ### Questions for students - Why does the current not disappear immediately after disconnecting the source? - Why is energy stored in the magnetic field? - How is that energy later dissipated as heat? - Why can overvoltage appear when a circuit is suddenly opened? ### Planned animation An animation should show current build-up and decay in an RL circuit together with magnetic energy and the induced voltage across the coil. ### Interactive app for this chapter We now use a dedicated HTML animation for RL self-induction and decay: `Lecture_Notes/Physics/html_anim/electromagnetism_2/rl_decay_self_induction.html` The goal of this app is to show both phases of the problem in one place: - while the source is connected, the current builds up gradually, - the coil opposes the growth of current through self-induced EMF, - after disconnection, the source is removed but the current does not vanish instantly, - the coil reverses its voltage and keeps the current flowing, - the magnetic energy stored in the field decays into Joule heat. What the student should be able to read directly from the app: - the steady current is $I_0 = U/R$, - the time scale is $\tau = L/R$, - charging and decay are both exponential processes, - the sign of the coil voltage changes between the two phases, - the stored energy is $W = \frac12 L I^2$ and decreases during decay. The app currently lets us control: - inductance $L$, - resistance $R$, - source voltage $U$, - animation speed. This app should be embedded directly here: <iframe src="html_anim/electromagnetism_2/rl_decay_self_induction.html" width="100%" height="1240px" frameborder="0"></iframe> [html_anim/electromagnetism_2/rl_decay_self_induction.html](html_anim/electromagnetism_2/rl_decay_self_induction.html) ## 9. Ideal and Real Transformer This chapter is based on Problem 9 and should collect the induction ideas into one practical device. ### Main ideas - Alternating current in the primary coil creates changing magnetic flux in the core. - That changing flux induces voltage in the secondary coil. - In the ideal picture: $$ \frac{U_2}{U_1} = \frac{N_2}{N_1} $$ - Real transformers have losses: - resistive heating, - eddy currents, - hysteresis, - leakage flux. ### Questions for students - Where does the secondary voltage actually come from? - Why does a transformer require changing flux? - What makes a real transformer different from an ideal one? - Why are laminated cores used? ### Planned animation An animation should show the primary AC current, flux in the core, and the induced secondary voltage, with options to compare ideal and lossy behavior. ## 10. Eddy Currents and Contactless Braking This final chapter follows Problem 10 and closes the lecture with a vivid engineering application. ### Main ideas - Changing magnetic flux in a bulk conductor induces circulating currents. - These eddy currents generate magnetic effects that oppose the motion or the flux change. - The result can be contactless braking. ### Questions for students - Why do eddy currents appear in a solid conductor even without a wire loop? - Which materials enhance the effect? - How can the braking force be increased? - What are the advantages and disadvantages of eddy-current brakes? ### Planned animation An animation should show a metal plate moving through a magnetic field, the induced current loops, and the resulting braking effect. ## Final Summary The full lecture can be read as one coherent story: 1. A magnetic field bends the motion of charges. 2. Electric and magnetic forces can balance to select velocity. 3. Currents in loops behave like magnetic dipoles. 4. Changing flux induces EMF. 5. Motion in a field can generate voltage directly. 6. Induced currents create forces that oppose motion. 7. Electromagnetic systems convert mechanical energy into electrical energy and heat. 8. The same principles explain generators, transformers, and magnetic brakes. ## Suggested development order for animations If we want the lecture to become highly visual as quickly as possible, a good order is: 1. Lorentz force 2. Velocity selector 3. Rotating loop 4. Moving rod 5. Sliding rod on rails 6. Loop entering a field region 7. RL self-induction 8. Transformer 9. Eddy currents 10. Magnetic moment of a loop