Charged-Particle Motion in a Uniform Magnetic Field – Comprehensive Notes

Chapter 1 – Uniform Circular Motion of a Charged Particle in a Magnetic Field

When a charged particle with velocity $\vec v$ enters a region of uniform magnetic field $\vec B$ such that $\vec v \perp \vec B$ :
- Magnetic force: $\vec F_B = q\,\vec v \times \vec B$
- Magnitude: $F_B = qvB \sin\theta$
  - $\theta = 90^\circ \Rightarrow \sin\theta = 1 \Rightarrow F_B = qvB$ (maximum force)
  - $\theta = 0^\circ \text{ or } 180^\circ \Rightarrow F_B = 0$ (no deflection; particle moves undeviated)
- Direction by right-hand (palm) rule; for negative charge reverse the direction.

Centripetal Force Condition

Magnetic force provides the required centripetal force for circular motion: $qvB = \frac{mv^2}{r}$
- Rearranged for radius:
  $r = \frac{mv}{qB}$
- Substitute momentum $p = mv$ to obtain an alternative form:
  $r = \frac{p}{qB}$

Time Period (Time of Flight for One Revolution)

Distance traveled in one full circle: $2\pi r$
Time period $T = \dfrac{\text{distance}}{\text{speed}} = \dfrac{2\pi r}{v}$
Insert $r = \dfrac{mv}{qB}$ :
$T = \frac{2\pi (mv/qB)}{v} = \boxed{T = \frac{2\pi m}{qB}}$

Frequency

Definition: number of revolutions per second $f = \dfrac{1}{T}$
Using the expression for $T$ :
$f = \boxed{f = \frac{qB}{2\pi m}}$

Angular Frequency / Angular Velocity $\omega$

Two equivalent relationships used in exams:
- $\omega = 2\pi f$
- $\omega = \dfrac{2\pi}{T}$
Substitute either $f$ or $T$ to get:
$\boxed{\omega = \frac{qB}{m}}$
Key observation: $\omega$ , $f$ , and $T$ depend only on the charge-to-mass ratio $\frac{q}{m}$ and the magnetic field $B$ , not on the particle’s speed (provided motion is perpendicular to $B$ ).

Chapter 2 – Charges at Rest or Moving Parallel/Antiparallel to $\vec B$

Charge at rest ( $v = 0$ ): $F_B = 0$ → No acceleration; particle continues with zero velocity.
Charge moving parallel or antiparallel to $\vec B$ ( $\theta = 0^\circ, 180^\circ$ ): $F_B = 0$ → Particle travels straight, undisturbed.

Chapter 3 – Proton vs. Alpha-Particle Properties

Proton (symbol $p$ )
- Mass: $m_p = m$ (reference symbol used in the lecture)
- Charge: $q_p = q$
Alpha particle ( $\alpha$ ) = helium nucleus ( $^{4}\text{He}^{2+}$ )
- Composition: 2 protons + 2 neutrons
- Mass: $m_\alpha \approx 4m$ (neutron and proton masses almost equal)
- Charge: $q_\alpha = 2q$ (only the two protons contribute; neutrons are neutral)

Chapter 4 – Sample Problem: Equal-Radius Paths for Proton and Alpha Particle

Situation:
- Uniform $\vec B$ field (same region & magnitude for both particles)
- Both enter with velocities perpendicular to $\vec B$ (ensures circular motion)
- Radii of the two circular paths are given to be equal: $rp = r\alpha$

Using the Radius–Momentum Relation

$r = \frac{p}{qB}$

Set the radii equal and cancel the common $B$ :
$\frac{pp}{qpB} = \frac{p\alpha}{q\alpha B} \;\Longrightarrow\; \frac{pp}{p\alpha} = \frac{qp}{q\alpha}$
Insert charges:
$qp = q, \; q\alpha = 2q \Rightarrow \frac{pp}{p\alpha} = \frac{q}{2q} = \boxed{\tfrac{1}{2}}$
Common exam pitfall (highlighted in the lecture): do not divide by the mass ratio; the formula already involves momentum, not directly the mass.

Chapter 5 – Summary of Key Equations

Magnetic force (magnitude): $F = qvB\sin\theta$
Centripetal condition for circular motion ((\theta = 90^\circ)):
$qvB = \dfrac{mv^2}{r}$
Radius: $r = \dfrac{mv}{qB} = \dfrac{p}{qB}$
Time period: $T = \dfrac{2\pi m}{qB}$
Frequency: $f = \dfrac{qB}{2\pi m}$
Angular frequency (cyclotron frequency): $\omega = \dfrac{qB}{m}$

Chapter 6 – Practical & Theoretical Implications

Because $T$ , $f$ , and $\omega$ depend only on $\frac{q}{m}$ and $B$ , all particles with the same charge-to-mass ratio execute circular motion with the same frequency—basis of the cyclotron accelerator.
Knowing any one variable (radius, time period, frequency, or angular frequency) lets you derive the others using the boxed formulas.
Conceptual checkpoints tested in boards & entrance exams:
1. Zero magnetic force for $\vec v \parallel \vec B$ .
2. Maximum magnetic force for $\vec v \perp \vec B$ .
3. Derivation of $r$ , $T$ , $f$ , $\omega$ from $qvB = \dfrac{mv^2}{r}$ .
4. Comparative questions (e.g.
- proton vs. alpha particle radii, momenta, frequencies).
1. Right-hand palm rule direction questions, especially sign reversal for negative charges.

Chapter 1 – How a Charged Particle Moves in a Magnetic Field

When a charged particle (like a tiny electron or proton) with speed v enters a uniform magnetic field B (an area with an invisible magnetic push/pull) so that its path is straight across the field lines (vis perpendicular to B):
- Magnetic force (_F_B): This is the push or pull the magnetic field puts on the moving particle.
  - Its strength (how big the push/pull is): $F_B = qvB\sin\theta$
  - When the particle moves straight across the field ( $\theta = 90^\circ$ ), the force is strongest: $F_B = qvB$
  - If the particle moves along the field lines ( $\theta = 0^\circ$ or $180^\circ$ ), there's no force: $F_B = 0$ (it just keeps going straight).
- Direction of force: Use the right-hand (palm) rule. If the particle has a negative charge (like an electron), flip the direction you found.

Why it Moves in a Circle (Centripetal Force)

The magnetic force acts like a string, constantly pulling the particle towards the center, making it move in a perfect circle. This "pull-to-center" force is called centripetal force.
- The magnetic force ( $qvB$ ) provides the needed centripetal force ( $\frac{mv^2}{r}$ ): $qvB = \frac{mv^2}{r}$
We can use this to find the radius (r) of the circle the particle makes:

$r = \frac{mv}{qB}$

Since momentum (p) is mass (m) times velocity (v) ( $p = mv$ ), we can also write the radius as:

$r = \frac{p}{qB}$

Time for One Full Circle (Time Period)

The distance the particle travels in one full circle is the circumference ( $2\pi r$ ).
The time it takes to complete one circle (Time period T) is:

$T = \frac{\text{distance}}{\text{speed}} = \frac{2\pi r}{v}$

If we put in the formula for r:

$T = \frac{2\pi (mv/qB)}{v} = \boxed{T = \frac{2\pi m}{qB}}$

How Many Circles Per Second (Frequency)

Frequency (f) is how many full circles the particle makes in one second. It's the opposite of the time period: $f = \frac{1}{T}$
Using the formula for T:

$f = \boxed{f = \frac{qB}{2\pi m}}$

How Fast it Spins (Angular Frequency / Angular Velocity $\omega$ )

Angular frequency ( $\omega$ ) tells us how fast the particle is spinning in its circle.
- It's related to frequency and time period by: $\omega = 2\pi f$ or $\omega = \frac{2\pi}{T}$
Using the formula for f or T:

$\boxed{\omega = \frac{qB}{m}}$

Key thing to notice: $\omega$ , f, and T depend only on the particle's charge-to-mass ratio ( $\frac{q}{m}$ ) and the magnetic field ( $B$ ). They don't depend on how fast the particle is actually moving (its speed v), as long as it's moving straight across the magnetic field.

Chapter 2 – When the Magnetic Field Does Nothing

If the charged particle is not moving ( $v = 0$ ): There's no magnetic force ( $F_B = 0$ ). So, it just stays still.
If the charged particle moves exactly along the magnetic field lines (either with them or directly against them, $\theta = 0^\circ, 180^\circ$ ): There's no magnetic force ( $F_B = 0$ ). The particle just keeps moving in a straight line, unaffected.

Chapter 3 – Understanding Protons and Alpha Particles

Proton (symbol $p$ ):
- Mass: Let's call its mass m.
- Charge: Let's call its charge q (it's a positive charge).
Alpha particle ( $\alpha$ ) (this is the core of a Helium atom, $^{4}\text{He}^{2+}$ ):
- What it's made of: 2 protons + 2 neutrons.
- Mass: Since neutrons and protons have almost the same mass, an alpha particle is roughly 4 times heavier than a proton ( $m_\alpha \approx 4m$ ).
- Charge: It has a charge of +2 because of its two protons (neutrons have no charge) ( $q_\alpha = 2q$ ).

Chapter 4 – Solving a Problem: Proton vs. Alpha Particle Paths with Same Radius

The situation: Imagine a uniform magnetic field ( $\vec B$ ) that's the same everywhere. A proton and an alpha particle both enter this field, moving straight across the field lines, and they end up making circles of the exact same size ( $rp = r\alpha$ ).

Using the Radius-Momentum Formula

Our formula for radius is: $r = \frac{p}{qB}$
Since the radii are equal and the magnetic field B is the same for both, we can set them equal and cancel out B:

$\frac{pp}{qpB} = \frac{p\alpha}{q\alpha B}$

This simplifies to: $\frac{pp}{p\alpha} = \frac{qp}{q\alpha}$
Now, put in their charges (proton charge $qp = q$ , alpha particle charge $q\alpha = 2q$ ):

$\frac{pp}{p\alpha} = \frac{q}{2q} = \boxed{\tfrac{1}{2}}$

Important tip: This means the proton's momentum ( $pp$ ) must be half of the alpha particle's momentum ( $p\alpha$ ) for them to have the same radius. Don't get confused and divide by mass ratios directly here; the formula already uses momentum.

Chapter 5 – Quick Look at the Main Formulas

Magnetic force (strength): $F = qvB\sin\theta$
For circular motion (force is perpendicular): $qvB = \dfrac{mv^2}{r}$
Radius of the circle: $r = \dfrac{mv}{qB} = \dfrac{p}{qB}$
Time for one circle: $T = \dfrac{2\pi m}{qB}$
Revolutions per second: $f = \dfrac{qB}{2\pi m}$
How fast it spins: $\omega = \dfrac{qB}{m}$

Chapter 6 – Why This Is Important

Because T, f, and $\omega$ depend only on the charge-to-mass ratio ( $\frac{q}{m}$ ) and the magnetic field ( $B$ ), any particles with the same $\frac{q}{m}$ will spin in circles at the same frequency. This idea is used in machines like the cyclotron accelerator, which speeds up particles.
If you know just one of these values (like the radius, time period, frequency, or angular frequency), you can use the formulas to calculate all the others.
Things to remember for tests:
1. When is there no magnetic force on a charged particle? (When it moves parallel to the field).
2. When is the magnetic force strongest? (When it moves perpendicular to the field).
3. How to get the formulas for r, T, f, and $\omega$ starting from the main force equation ( $qvB = \dfrac{mv^2}{r}$ ).
4. Questions comparing different particles (like asking about the radii, momenta, or frequencies of a proton vs. an alpha particle).
5. How to use the right-hand palm rule to find the direction of the force, and remember to flip it for negative charges.