Chapter 25: Motion of Charged Particles

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement not replaced the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Welcome to the deep dive.

You know, when we try to picture understanding the universe, I feel like we always jump straight to massive telescopes.

Oh yeah.

Galaxies, black holes, that sort of grand scale stuff.

Right.

But honestly, some of the most profound discoveries we've ever made happen when we look at things that are just impossibly small.

Things completely invisible to the naked eye.

Exactly.

So today, our mission is a special one.

We're doing a one -on -one tutoring session specifically covering chapter 25 of the Cambridge International AS and A level physics course book.

Motion of charged particles.

That's the one.

And we are going to unpack this chapter in the exact order it appears.

We're breaking down the physical principles, the derivations, the equations.

So that you don't just, you know, blindly plug numbers into a calculator.

Right.

We want you to actually understand why the physics works.

So think about a photograph from a bubble chamber.

Those are beautiful.

It's this visual of subatomic particles leaving these gently curving tracks or sometimes tightly wound spirals.

Yeah.

And what actually pauses those beautiful curves?

Well, it's magnetic fields.

I mean, understanding that invisible interaction is what allows us to literally control the building blocks of the universe.

Which is wild to think about.

So let's start with section 25 .1, seeing the invisible.

How do we actually observe this force?

We use something called an electron beam tube.

Imagine a glass bulb, right?

But with all the air completely sucked out.

So a total vacuum.

Exactly.

And at one end you have a heated cathode, which basically releases this little cloud of electrons.

They just sort of boil off the metal because it's so hot, right?

Yep.

Thermionic emission.

And then there's a positively charged anode nearby that just violently pulls those negative electrons forward.

Because opposites attract.

Right.

And it pulls them through a tiny hole, which forms them into this perfectly straight narrow beam.

Okay.

So we have a straight beam of electrons shooting through a vacuum.

What happens when we bring a magnet near it?

The beam bends.

You can physically see it curve.

Because the electrons are experiencing a magnetic force.

Exactly.

But if you want to predict which way that beam is going to bend, you have to use Fleming's left -hand rule.

And there is a massive trap here.

Oh, right.

Remember the left -hand rule.

First finger is the magnetic field, second finger is the current, and your thumb is the force.

That's the one.

But think about it.

The rule is based on conventional current.

Wait.

Conventional current is the flow of positive charge, right?

Yes.

But electrons are negatively charged.

Ah.

Okay.

I see the trap.

Yeah.

So if your actual physical electrons are moving, say left to right, your conventional current is technically flowing from right to left.

It's completely backward.

Well, entirely backward.

So when you hold up your left hand for the exam and you point your second finger, you have to point it in the exact opposite direction to the electron beam.

That is such a crucial tip.

So if the beam goes left, my current finger points right.

Exactly.

And your thumb shows the force.

And notice the geometry of your hand there.

That magnetic force is always acting at exactly 90 degrees to the velocity of the electrons.

It's always a perpendicular push.

Okay.

So what's the actual math for that push?

The force, which is F, depends on three things.

The magnetic flux density, B, the particle's charge, Q, and its speed, V.

So F equals B times Q times V, F equals BQV.

Right.

And if it hits the field at an angle, say angle theta, you only care about the perpendicular part.

So it becomes F equals BQV sine theta.

Okay.

Let's pause and unpack this because I need a why moment here.

Let's take it.

Well, we know the force on a regular physical wire is F equals BIL, magnetic field, current, and length.

How does a macroscopic wire formula turn into a microscopic particle formula?

This is one of my favorite derivations.

Okay.

So think about what current I actually is.

It's the rate of flow of charge.

So charge divided by time, Q over T.

Perfect.

So substitute Q over T for the I and F equals BIL.

Okay.

So now I have F equals B times Q over T times L.

Right.

Now look at the L over T part, length divided by time.

If a particle travels the length, L and a time T.

Wait, distance over time is just speed.

It's velocity.

There you go.

So the L over T just becomes V.

Wow.

So F equals BIL literally just rearranges into F equals BQV.

Exactly.

The macroscopic force on the wire is literally just the sum of billions of tiny BQV forces on individual electrons.

That makes so much sense.

Okay.

So let's move to section 25 .2.

Going in circles.

Yeah.

The fine beam tube.

Right.

So imagine we shoot our electron beam vertically into a spherical glass tube, but this time there's a uniform horizontal magnetic field across the whole thing.

And there's a little bit of low pressure gas inside, right?

Yeah.

So when the electrons hit the gas, they glow.

And we see the beam bend into a perfect continuous circle.

But why a circle?

Well, remember that perpendicular push we talked about?

Yeah.

The force is always exactly 90 degrees to the velocity.

Right.

And if a force is always pushing you exactly sideways, it can't speed you up or slow you down.

It does zero work in your direction of travel.

It can only change your direction.

Exactly.

And a constant force that only changes your direction.

That's a centripetal force.

Ah.

So the magnetic force is acting as the centripetal force.

Precisely.

So we can equate them.

The magnetic force, bqv, equals the centripetal force, which is mv squared over r.

Okay.

bqv equals mv squared over r.

I can cancel one of the v's on both sides.

Yep.

And if you rearrange that to solve for the radius, r, you get r equals mv divided by bq.

So what does that actually mean for the particle?

The top of the fraction is mass times velocity.

That's momentum.

Right.

mv is momentum.

Or heavier and faster particles have more momentum.

They're like a massive, fast -moving truck trying to take a corner.

They're going to take a much wider turn, so a larger radius.

Yeah.

But the bottom of the fraction is b times q, magnetic field and charge.

That's like the grip of the tires on the road, right?

That's a great analogy.

Yeah.

A stronger magnetic field or a higher charge pulls them in for a much tighter, smaller turn.

But wait, practically speaking, I can measure the radius of that glowing circle with a ruler.

And I know the magnetic field strength I applied, but I can't exactly point a police radar gun at an electron to get its speed, v.

So how do we actually figure out how fast it's going?

We use the law of conservation of energy.

Think about how we got the electron moving in the first place.

The electron gun.

We used an accelerating voltage to pull it.

Exactly.

So the electrical work done on the electron is its charge, little e times the accelerating voltage, v.

Okay, so e times v.

And all of that electrical work is converted entirely into kinetic energy.

Which is one half mv squared.

Yes.

So e times v equals one half mv squared.

You know the voltage you dialed in on the machine.

You know the mass and charge.

So you can just solve for the speed, v.

Oh, that is brilliant.

We use the wall plug voltage to find the speed of an invisible particle.

It really is elegant.

And that calculation is actually essential for finding the specific charge of an electron, which is its charge to mass ratio, e over m.

Okay, so that's a single clean beam.

But what if things are messy?

Moving on to section 25 .3.

Ah, velocity selection.

The ultimate sorting hat.

Imagine a mass spectrometer where you have a beam of mixed particles all flying at completely different speeds.

And we only want the ones going one exact speed.

To do that, we create a velocity selector.

We pass the beam through a narrow slit between two parallel horizontal plates.

One plate is positive, one is negative.

So we have a constant downward electric field.

Exactly.

And the electric force pulling down on the particles is e times q.

Electric field strength times charge.

And the key here is that the electric force does not care about speed.

A slow particle and a fast particle feel the exact same downward pull.

Right.

But in that exact same space, we apply a magnetic field.

And we set it up so the magnetic force pushes upwards.

Yep.

And what's the formula for magnetic force again?

VQV.

And because it has a V in it, the magnetic force heavily depends on speed.

Exactly.

So you've got this battle of the forces.

The downward electric force is perfectly constant.

But the upward magnetic force changes depending on the particle's speed.

It's like an exclusive bouncer at a club.

If you're running in too fast, the upward magnetic force becomes huge and it slams you into the ceiling.

And if you're moving too slow, the electric force wins and you get dragged down into the floor.

Right.

But there's a sweet spot.

A specific speed where the constant downward electric force exactly equals the upward magnetic force.

So EQ equals BQV.

Yes.

And look at that equation.

Notice anything.

The Q is on both sides.

So the charge completely cancels out.

Exactly.

It doesn't matter what the particle's charge is.

The mass doesn't matter either.

If you cancel the Q, you're left with E equals BV.

Or V equals E divided by B.

So only particles moving at exactly the speed of E divided by B will fly perfectly straight through the slit at the end.

It's pure elegant filtering.

Okay.

But this whole thing requires us to know the exact strength of the magnetic field B.

How do we actually measure that in a lab?

That brings us to section 25 .4, the Hall Effect.

Right.

Magnetic fields creating voltage.

So to measure a magnetic field, we use a Hall probe.

Imagine a flat rectangular slice of a conductor.

And we run a steady current from one end of the slice to the other.

Okay, so I've got a flat rectangle with electrons flowing straight through it.

Now place that slice directly into a perpendicular magnetic field.

What happens to those moving electrons?

Well, they're moving charges in a magnetic field, so they get pushed sideways.

Exactly.

The magnetic field physically shoves the flowing electrons toward one specific edge of the rectangle.

So you get a traffic jam of negative electrons bunched up on one edge.

Which leaves the opposite edge with a lack of electrons, making it positive.

So one side is negative, the other side is positive.

That means there's a potential difference across the slice.

Yes.

And we call that the Hall Voltage, VH.

But wait, if they keep getting pushed to the edge, won't they just keep piling up forever?

Ah, well that's step two.

That buildup of charge creates its own internal electric field across the slice.

Because you have a positive side and a negative side.

Exactly.

And that internal electric field exerts an electric force that pushes the electrons back toward the middle.

Oh, it's another tug of war.

The magnetic field pushes them to the edge, but the new electric field pushes them back.

Yes.

And equilibrium is reached when the electric force pushing back exactly equals the magnetic force pushing sideways.

Okay, let's do the math for that.

The electric force is the charge of an electron, little e times the electric field e.

And the magnetic force is b times e times v.

So e times e equals b times e times v.

The little e's cancel out.

Leading us with the electric field e equals b times v.

But e is also voltage divided by distance, right?

So the hall voltage, vh, divided by the width of the slice d.

Exactly, substitute that in.

Vh over d equals b.

Which means the hall voltage vh equals b times v times d.

We're almost there.

Now, remember the drift velocity equation from the electricity chapter?

Current i equals n times a times v times e.

Right.

N is the number of density -free electrons, a is the cross -sectional area, v is the drift velocity and e is the elementary charge.

So rearrange that for v.

You get v equals i over an a.

Remember the area a is just the thickness of the slice, t times its width d.

Oh wow, okay.

If we substitute all of that back into our hall voltage equation, things are going to cancel out.

They do.

And you eventually get the master equation.

The hall voltage vh equals b times i divided by n times t times q.

Let me make sure I've got that.

Hall voltage equals magnetic field times current over a number density times thickness times charge.

You got it.

That is the hall voltage.

Let me push back on the design of this probe, though.

Go for it.

I want a probe that gives me a massive, easy -to -read voltage.

So why wouldn't I just make the slice out of a highly conductive metal, like copper?

It sounds logical, but it's a brilliant twist of physics.

Look at the master equation.

Where is the number density, Nan?

It's on the bottom of the fraction.

It's inversely proportional.

Right.

And copper has a massive number density.

It is absolutely packed with free electrons.

Ah, like a massively crowded hallway.

If I need a hundred people to pass through a hallway every minute, and it's packed shoulder to shoulder, they don't have to run, they just slowly shuffle forward.

Exactly.

The electrons in copper move with an incredibly slow drift velocity.

And since the magnetic force depends on speed?

The magnetic push sideways is incredibly weak, so the hall voltage would be tiny.

Too tiny to measure easily.

That's why we don't use metals, we use semiconductors.

Because semiconductors have thousands of times fewer charge carriers.

So it's an empty hallway.

Exactly.

To maintain the same current, those few electrons have to absolutely sprint.

Higher speed means a massive magnetic push, which gives a much larger, easily readable hall voltage.

That is such a clever engineering trick.

It is.

Physics in action.

Okay, bringing it all home with section 25 .5, history in the making,

discovering the electron.

Yeah, these aren't just equations.

This exact math changed our understanding of reality.

Take us back to 1897, J .J.

Thompson.

Right, so back then scientists thought the atom was the smallest thing in existence.

The indivisible building block.

But Thompson was playing around with deflection tubes, which are basically the electron beam tubes we talked about at the start.

He was studying cathode rays.

He noticed they were deflected by electric and magnetic fields, which proved they were negatively charged particles.

But how did he figure out what they actually were?

He used the exact cross -field technique we talked about with the velocity selector.

The tug -of -war.

He balanced the electric force and the magnetic force until the beam went perfectly straight.

Yes.

And by measuring the field strengths needed to balance the beam, he was able to calculate the charge -to -mass ratio,

the specific charge.

He didn't know the exact charge or the exact mass yet, right?

No, but the ratio itself was enough.

It was staggering.

It proved that these particles were thousands of times lighter than a hydrogen atom.

The lightest known atom at the time.

Exactly.

Thompson had discovered the electron.

He shattered the idea that the atom was indivisible.

That is incredible.

And then later, Robert Milliken does his famous oil drop experiment.

Right.

Milliken suspends tiny charged oil droplets in an electric field to find the exact elemental charge.

And once we had Milliken's exact charge and Thompson's charge -to -mass ratio, it's It was basic division to find the exact mass of a single electron.

Humanity finally had the mass of the fundamental building block of the universe.

Wow.

We have really covered some ground today.

From basic forces bending a beam to calculating perfect orbital circles.

To using crossed fields to select exact velocities.

And then using the Hall effect and semiconductors to measure the invisible fields themselves.

It's so interconnected.

It really is.

And you know, just to leave you with a thought,

look at question 10 in the chapter.

Oh, the exam question.

Yeah.

Think about the bigger picture.

Imagine protons and helium nuclei just violently exploding out of the sun, hitting Earth's atmosphere at a million meters per second.

Just a barrage of charged cosmic particles.

Right.

And the Earth has a massive magnetic field.

It acts exactly like the fields in our equations.

It pushes them perpendicularly.

Yes.

It captures these high speed particles in massive tightly wound orbital arcs, funneling them down over the north and south poles.

Wait, is that what causes the auroras, the northern lights?

Exactly.

When they hit our atmosphere, they glow.

Just like the gas in our little fine beam tube experiment.

The exact same math happening in a tiny vacuum tube governs the magnificent auroras in the sky.

That is the true beauty of physics.

That is unbelievable.

While you've tackled some serious physics today, take a breath, definitely review those derivations we talked through, and I promise you will absolutely crush this topic.

On behalf of the Last Minute Lecture team, a warm thank you for learning with us.

Keep questioning, keep calculating, and we'll catch you on the next Deep Dive.