Physics | Motion in electromagnetic fields

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Hello, Future Physicist! Navigating Motion in Electromagnetic Fields (Topic D.3)

Welcome to one of the most exciting topics in the Fields section! Here, we stop looking at static charges and start looking at what happens when charges *move* through electric and magnetic fields. This is the core principle behind everything from old TV screens (Cathode Ray Tubes) to massive particle accelerators and even how we measure the masses of atoms.

Don't worry if this seems tricky at first. It’s all about direction and balance. We’ll break down the forces piece by piece.

Quick Prerequisite Check: Before diving in, ensure you remember:

• Electric fields exert a force \(F_E = qE\), parallel to the E-field lines (for positive charges).
• Magnetic fields (B-fields) are measured in Teslas (T).

1. The Lorentz Force: The Total Force on a Moving Charge (SL & HL)

When a charged particle (like an electron or a proton) moves through a region where both electric and magnetic fields exist, it experiences a combined force known as the Lorentz Force.

1.1 Force Due to the Electric Field (\(F_E\))

This is the familiar force from Topic D.2. It acts along the direction of the electric field lines (for a positive charge, \(+q\)).

\[F_E = qE\]

• The electric force is always parallel (or anti-parallel) to the electric field vector \(E\).
• This force can do work on the particle, changing its speed.

1.2 Force Due to the Magnetic Field (\(F_B\))

This force is special because it only acts on a charge if it is moving relative to the magnetic field. If the charge is stationary (\(v=0\)), there is no magnetic force!

The magnitude of the magnetic force is:

\[F_B = qvB \sin\theta\]

Where:

• \(q\) is the charge magnitude (C).
• \(v\) is the velocity (m s\(^{-1}\)).
• \(B\) is the magnetic field strength (T).
• \(\theta\) is the angle between the velocity vector (\(v\)) and the magnetic field vector (\(B\)).

Key Takeaway for Magnetic Force:

• If the particle moves parallel or anti-parallel to the B-field (\(\theta = 0^\circ\) or \(180^\circ\)), \(\sin\theta = 0\), so \(F_B = 0\).
• The force is maximum when the particle moves perpendicular to the B-field (\(\theta = 90^\circ\)), so \(F_B = qvB\).

1.3 Determining the Direction (The Right-Hand Rule - RHR)

Since magnetic force involves direction, we use the Right-Hand Rule (RHR, sometimes called Fleming's Left-Hand Rule in engineering, but RHR is standard for Physics vectors).

The IB Physics Right-Hand Rule (for the vector cross product):

1. Point your fingers in the direction of the velocity (\(v\)).
2. Curl your fingers towards the direction of the magnetic field (\(B\)).
3. Your thumb points in the direction of the force (\(F\)) for a POSITIVE charge (\(+q\)).

If the charge is NEGATIVE (\(-q\)) (like an electron), the force direction is exactly opposite to your thumb.

Analogy: Imagine you are trying to push the B-field (B) with your fingers so it lines up with your velocity (v). The resulting shove (F) comes from your thumb.

2. Motion of a Charge in a Uniform Magnetic Field (SL & HL)

Let's simplify and assume the electric field \(E\) is zero, and the charge moves *perpendicularly* to a uniform magnetic field \(B\).

2.1 Circular Motion

Since \(F_B\) is always perpendicular to \(v\), the force acts toward the center of a circle. This means the magnetic force acts as the centripetal force (\(F_c\)).

Step-by-Step Derivation:

1. The net force is the magnetic force (since \(E=0\)): \(F_{net} = F_B = qvB\). (Since \(\theta = 90^\circ\)).
2. For circular motion, the required centripetal force is: \(F_c = \frac{mv^2}{r}\).
3. Set them equal: \(F_B = F_c\)

\[qvB = \frac{mv^2}{r}\]

We can solve this for the radius (\(r\)) of the circular path:

\[r = \frac{mv}{qB}\]

This is a crucial IB formula!

Common Mistakes to Avoid:

• If the particle enters the field at an angle *not* perpendicular to B, the motion will be a helix (a spiral path), not a flat circle.
• Remember that the direction of rotation depends on the sign of the charge (\(q\))—positive charges curve one way, negative charges curve the other (RHR!).

2.2 Period and Frequency of Circular Motion

We can also calculate the time it takes for the particle to complete one full circle (the period, \(T\)).

We know that speed \(v = \frac{2\pi r}{T}\). Substituting the radius equation \(r = \frac{mv}{qB}\):

\[v = \frac{2\pi}{T} \left( \frac{mv}{qB} \right)\]

Notice that the velocity (\(v\)) cancels out!

Solving for the Period (\(T\)):

\[T = \frac{2\pi m}{qB}\]

The Frequency (\(f\)) is just the inverse of the period (\(f = 1/T\)):

\[f = \frac{qB}{2\pi m}\]

3. The Velocity Selector (SL & HL)

How do scientists ensure that only particles moving at a specific, desired speed are allowed into their detector or experiment? They use a Velocity Selector.

3.1 Principle of the Velocity Selector

A velocity selector utilizes crossed fields—a region where a uniform electric field (\(E\)) and a uniform magnetic field (\(B\)) are set up perpendicular to each other, and also perpendicular to the particle’s initial path (\(v\)).

Imagine a positively charged particle entering the selector. The forces acting on it are \(F_E\) and \(F_B\).

• The electric field pulls the particle in one direction (e.g., up).
• The magnetic field pulls the particle in the opposite direction (e.g., down, using RHR).

If the particle moves straight through without deflection, the two opposing forces must be perfectly balanced:

\[F_E = F_B\]

3.2 The Selected Velocity

Substitute the formulas for the forces:

\[qE = qvB\]

The charge \(q\) cancels out (meaning the selected speed is the same for all particles, regardless of charge or mass!), leaving the formula for the velocity \(v\) that passes straight through:

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

Memory Aid: Think of the equation as the Sleek Speed Selector, where the desired speed \(v\) is simply the ratio of the two fields, \(E\) over \(B\).

If the particle is too slow: \(F_B\) is too small. \(F_E\) dominates, and the particle is deflected in the E-field direction.

If the particle is too fast: \(F_B\) is too large. The magnetic force dominates, and the particle is deflected in the B-field force direction.

Only particles with \(v = E/B\) pass through undeflected.

Summary and Key Takeaways

This chapter taught us how electric and magnetic fields dictate the paths of moving charges.

• The Lorentz Force is the combination of the electric force (\(F_E\)) and the magnetic force (\(F_B\)).
• The magnetic force (\(F_B\)) is unique because it is always perpendicular to velocity, meaning it changes direction but not speed.
• This perpendicular force causes charges moving through a uniform B-field to follow a circular path. The radius of this circle is directly proportional to the momentum (\(mv\)).
• The Velocity Selector uses balanced electric and magnetic forces to isolate particles moving at a specific speed \(v=E/B\).

You now have the tools to understand the dynamics of charge movement—a fundamental concept in modern physics and technology!

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