Study Notes: Moving Charges in a Magnetic Field (9630 A-Level Physics)
Hello future physicist! This chapter is where we move beyond stationary charges and currents in wires, and look at the fascinating physics of how individual charged particles behave when they zoom through a magnetic field. This knowledge is crucial for understanding everything from particle accelerators to the aurora borealis!
Don't worry if magnetism feels a little abstract. We'll use simple rules and geometry to visualize the forces involved. Let's get started!
1. The Magnetic Force on a Single Moving Charge
Before tackling moving charges, let’s quickly recall what a magnetic field is. We use the term Magnetic Flux Density, \(B\), measured in Tesla (T).
The Conditions for a Force
When a charged particle (like an electron or a proton) moves through a magnetic field, it experiences a force. However, this force only occurs if two key conditions are met:
- The particle must be moving (if \(v = 0\), \(F = 0\)).
- The particle's velocity (\(v\)) must have a component perpendicular to the magnetic field (\(B\)).
Analogy: Imagine a boat trying to cross a fast-moving river (the magnetic field). If the boat moves directly downstream (parallel to the current), the river doesn't push it sideways. Only if it tries to cross the river (perpendicular motion) does it feel a strong sideways push.
The Magnetic Force Equation (\(F = BQv\))
The magnitude of the force \(F\) acting on a charged particle moving perpendicular to a uniform magnetic field \(B\) is given by the formula:
\(F = BQv\)
Where:
- \(F\) is the magnetic force (N)
- \(B\) is the magnetic flux density (T)
- \(Q\) is the magnitude of the charge (C)
- \(v\) is the velocity of the particle (m s-1)
Important Note: This formula is used when the velocity \(v\) is strictly perpendicular to the magnetic field \(B\). This is the key scenario required by the syllabus. If the particle moves parallel (or anti-parallel) to the field lines, the force is zero.
Quick Review Box: The Tesla (T)
The definition of the Tesla comes directly from this relationship (and the related \(F=BIL\)). One Tesla is the magnetic flux density that causes a force of 1 N on a 1 C charge moving at 1 m s-1 perpendicular to the field.
Key Takeaway: The magnetic force depends linearly on the field strength, the charge magnitude, and the speed. If you double the speed, you double the force.
2. Determining the Direction of the Force: Fleming's Left Hand Rule
Since force is a vector, we need a method to find its direction. We use Fleming's Left Hand Rule (LHR).
Step-by-Step Use of Fleming's LHR
Hold your left hand so that your thumb, forefinger, and middle finger are all mutually perpendicular (at 90 degrees to each other):
- Forefinger (Field, \(B\)): Points in the direction of the magnetic field (North to South).
- Middle Finger (Current/Charge Velocity, \(v\)): Points in the direction of the conventional current (or the velocity of a positive charge).
- Thumb (Force, \(F\)): Shows the direction of the resulting magnetic force.
Memory Aid (The FBI Mnemonic):
- Fumb (Force)
- Before finger (Field)
- Inside finger (Current/Charge movement)
Crucial Detail: Handling Negative Charges
The biggest mistake students make is applying LHR directly to electrons!
- Positive Charges (e.g., Protons): Use Fleming's LHR exactly as described. The middle finger points in the direction of \(v\).
-
Negative Charges (e.g., Electrons): Electrons move opposite to conventional current. Therefore, if the electron moves right, the effective current direction is left. You have two options:
- Apply Fleming's LHR using the direction opposite to the particle’s motion.
- Use the Left Hand Rule normally, pointing the middle finger in the direction of the electron's motion, but then reverse the resulting force direction shown by your thumb.
Did you know? The force on a charged particle is sometimes called the "Lorentz force."
Key Takeaway: Use Fleming's Left Hand Rule, but be extra careful to define the direction of the charge's motion: if it's negative, the force direction is flipped!
3. The Circular Path of Charges
This is where the physics gets really neat! If a charged particle enters a uniform magnetic field \(B\) perpendicularly, the magnetic force \(F\) is always at a 90-degree angle to the velocity \(v\).
Why it Moves in a Circle
When a force is constantly perpendicular to the direction of motion:
- The force does no work (\(W = Fs \cos \theta\), and here \(\theta = 90^\circ\)).
- Because no work is done, the particle's kinetic energy (\(E_k\)) remains constant, meaning its speed (\(v\)) does not change.
- The force acts as a centripetal force, constantly pulling the particle towards the center of a circle, changing its direction but not its speed.
Therefore, the path of a charged particle entering a uniform magnetic field perpendicularly is a circular path.
Calculating the Radius of the Circular Path (\(r\))
Since the magnetic force provides the necessary centripetal force, we can equate the two forces:
Force Magnetic = Force Centripetal
\(F_{magnetic} = F_{centripetal}\)
\(BQv = \frac{mv^2}{r}\)
We can rearrange this important equation to find the radius \(r\) of the path:
\(r = \frac{mv}{BQ}\)
- \(m\) is the mass of the particle (kg)
- \(v\) is the speed (m s-1)
- \(BQ\) is the charge multiplied by the field strength
Deep Dive: What the Radius Tells Us
The term \(mv\) is the momentum of the particle. The radius \(r\) is directly proportional to the momentum.
- High Speed or Mass (High Momentum): \(r\) is large. The particle travels in a wide circle.
- High Field Strength (\(B\)) or High Charge (\(Q\)): \(r\) is small. The particle is forced into a tighter circle.
Calculating the Period (\(T\)) and Frequency (\(f\))
The time taken for one full rotation (the period \(T\)) is the circumference divided by the speed:
\(T = \frac{2\pi r}{v}\)
Substitute the radius formula (\(r = \frac{mv}{BQ}\)) into the period equation:
\(T = \frac{2\pi}{v} \left( \frac{mv}{BQ} \right)\)
The \(v\) terms cancel out!
\(T = \frac{2\pi m}{BQ}\)
This is super neat: the period \(T\) is independent of the speed (\(v\)) and the radius (\(r\)). A fast particle travels in a larger circle, but it completes the circle in exactly the same amount of time as a slow particle in a tight circle (provided \(B\) and the particle's characteristics \(m, Q\) are the same).
Since frequency \(f = 1/T\):
\(f = \frac{BQ}{2\pi m}\)
Common Mistake to Avoid: Assuming the magnetic field changes the particle's speed. It only changes direction. Only an electric field can accelerate the particle (change its kinetic energy).
Key Takeaway: The magnetic force provides the centripetal force, resulting in a circular path where the period of rotation is independent of the particle's speed.
4. Application: The Cyclotron
The circular motion property, particularly the fact that the period is independent of speed, is the fundamental principle behind devices like the cyclotron, which is used to accelerate charged particles to very high speeds (and therefore high kinetic energy) for use in research or medical treatments.
How a Cyclotron Works
A cyclotron consists of two D-shaped, hollow metal containers (called "dees") placed face-to-face inside a strong, uniform magnetic field (\(B\)).
Step 1: Injection and Circulation
A charged particle (e.g., a proton) is injected near the centre. The magnetic field immediately forces it into a circular path inside one of the dees, following the law \(r = \frac{mv}{BQ}\).
Step 2: Acceleration (The Electric Field)
The gap between the dees has a high-frequency alternating electric field applied across it.
- When the particle crosses the gap, the electric field gives it a "kick," increasing its speed (\(v\)) and thus its kinetic energy (\(E_k\)).
Step 3: Increasing Radius
Because the speed \(v\) has increased, the radius \(r\) of the particle's path must also increase (\(r \propto v\)). The particle spirals outwards in a path of increasing radius.
Step 4: Synchronisation
Crucially, because the period \(T\) is constant (\(T = \frac{2\pi m}{BQ}\)), the time it takes the particle to complete each half-circle is the same, regardless of the radius.
- The frequency of the alternating electric field is carefully matched (or "tuned") to this constant orbital frequency (\(f\)).
- Every time the particle reaches the gap, the electric field reverses direction, ensuring the particle always gets a boost, constantly accelerating it.
Once the particle reaches the outer edge (the maximum radius), it is deflected out to hit a target. Cyclotrons are critical tools for producing radioactive tracers used in medical imaging.
Key Takeaway: The cyclotron exploits the fact that the time taken for a charge to complete a semi-circle in a uniform magnetic field is constant, allowing efficient synchronisation with an electric field to provide maximum energy gain.