Welcome to the World of Magnetic Forces!
Hello future Physicists! This chapter is where electricity (moving charges) meets magnetism in a spectacular way. Understanding the Force on a Moving Charge is crucial because it explains how motors work, how television screens display images, and how scientists guide particles in accelerators like CERN.
Don't worry if this topic feels a bit abstract. We'll break it down using simple rules and easy-to-remember concepts. By the end, you will be able to predict the path of any charged particle zooming through a magnetic field!
1. The Magnetic Force Equation
1.1 Calculating the Magnitude of the Force
When a charged particle moves through a magnetic field, it experiences a force. This magnetic force (\(F\)) depends on four key factors:
- The strength of the magnetic field (\(B\)).
- The magnitude of the charge (\(Q\)).
- The velocity of the charge (\(v\)).
- The angle (\(\theta\)) between the velocity vector and the magnetic field lines.
The mathematical relationship is given by the formula:
Magnetic Force \(F\): $$F = BQv \sin \theta$$
Key Definitions and Units:
- \(F\): Force (Newton, N)
- \(B\): Magnetic Flux Density (Tesla, T)
- \(Q\): Charge (Coulomb, C)
- \(v\): Velocity (metres per second, m/s)
- \(\theta\): Angle between \(v\) and \(B\) (degrees or radians)
Special Cases for the Angle (\(\theta\)):
The \(\sin \theta\) term is very important. It tells us when the force is maximum or zero:
- Maximum Force: When \(\theta = 90^\circ\) (\(\sin 90^\circ = 1\)). The charge moves perpendicular to the field lines. \(F_{max} = BQv\).
- Zero Force: When \(\theta = 0^\circ\) or \(180^\circ\) (\(\sin 0^\circ = 0\)). The charge moves parallel or anti-parallel to the field lines. If you shoot a proton along a magnetic field line, it experiences no magnetic force.
Quick Review: The force only acts on the component of velocity that is perpendicular to the magnetic field. If the charge is stationary (\(v=0\)), there is no magnetic force.
2. Determining the Direction of the Force
2.1 Using Fleming's Left-Hand Rule (LHR)
Since force is a vector, we must determine its direction. For a moving charge, we use a variation of the familiar Fleming's Left-Hand Rule (LHR). Remember that a moving positive charge is defined as conventional current.
The LHR Mnemonic: FBI
- Forefinger: Direction of the Field (\(B\))
- Base Finger (Middle): Direction of the Intentional current (Positive charge motion, \(Qv\))
- Thumb: Direction of the Thrust or Force (\(F\))
Step-by-Step for Determining Direction:
Case 1: Positive Charge (e.g., Proton)
If the charge \(Q\) is positive, the direction of the velocity (\(v\)) is the direction of the conventional current (\(I\)).
- Point your Forefinger in the direction of the magnetic field lines (North to South).
- Point your Middle Finger in the direction the positive charge is moving.
- Your Thumb will show the direction of the magnetic force exerted on the charge.
Case 2: Negative Charge (e.g., Electron)
If the charge \(Q\) is negative, the conventional current direction (\(I\)) is opposite to the direction the charge is actually moving.
Trick: Use the LHR as normal, pretending the electron is positive, but then reverse the final force direction (the thumb direction).
Did you know? The direction of the force is always perpendicular to both the velocity (\(v\)) and the magnetic field (\(B\)). This means the magnetic force can change the direction of motion, but never the speed or kinetic energy of the charge!
3. Charged Particle Motion in a Uniform Magnetic Field
3.1 Circular Motion (The Perpendicular Case)
When a charged particle enters a uniform magnetic field perpendicularly (\(\theta = 90^\circ\)), the resulting force \(F = BQv\) is:
- Constant in magnitude (since \(B, Q, v\) are constant).
- Always directed perpendicular to the velocity.
This perpendicular, constant force is exactly the condition required for Uniform Circular Motion.
Analogy: Imagine a hammer throw athlete spinning a weight on a chain. The tension force exerted by the chain is always directed inwards (perpendicular to the weight's velocity) causing it to move in a circle. The magnetic force acts like this tension!
Relating Forces and Finding the Radius (\(r\))
Since the magnetic force (\(F_B\)) provides the necessary centripetal force (\(F_c\)), we set them equal:
$$F_B = F_c$$ $$BQv = \frac{mv^2}{r}$$
We can rearrange this equation to find the radius of the circular path (\(r\)):
$$r = \frac{mv}{BQ}$$
This formula shows that:
- Faster particles (\(v\)) or heavier particles (\(m\)) will have a larger radius.
- Stronger fields (\(B\)) or larger charges (\(Q\)) will result in a smaller radius.
3.2 Helical Motion (The Non-Perpendicular Case)
If the charge enters the field at an angle other than 90° or 0°, the motion can be broken down into two components:
- \(v_{parallel}\) (along \(B\)): This component experiences no force, so the particle moves at a constant speed in that direction.
- \(v_{perpendicular}\) (perpendicular to \(B\)): This component creates the circular motion.
The combination of constant forward speed and circular motion creates a helical (spiral) path, like a spring or a corkscrew.
4. The Hall Effect and Hall Voltage
4.1 Origin of the Hall Voltage (\(V_H\))
When a current flows through a flat conductor (like a strip of metal), the charge carriers are moving. If this conductor is placed in a magnetic field (\(B\)) perpendicular to the current (\(I\)), the magnetic force pushes these moving charges to one side of the strip.
- The Build-up: As charges accumulate on one side, an Electric Field (\(E\)) is created across the width of the strip.
- Balance: This electric field exerts an opposing electric force (\(F_E\)) on the charges.
- Equilibrium: The charges stop migrating to the side when the magnetic force is exactly balanced by the electric force: $$F_{magnetic} = F_{electric}$$ $$BQv = EQ$$
- This electric field creates a measurable potential difference across the sides of the conductor, known as the Hall Voltage (\(V_H\)).
4.2 Deriving the Hall Voltage Formula (Required by Syllabus 20.3.3)
The syllabus requires us to derive and use the formula relating Hall voltage to the current, field, and material properties. Let's use the balanced force condition \(E = Bv\).
Step 1: Relate E to \(V_H\)
The electric field strength \(E\) inside the conductor is related to the potential difference \(V_H\) and the width \(w\) of the strip by \(V_H = Ew\).
Step 2: Substitute velocity \(v\)
We need to eliminate the drift velocity \(v\). Recall the current equation from earlier chapters:
$$I = Anvq$$
Where \(A\) is the cross-sectional area and \(n\) is the number density of charge carriers. If the thickness of the strip is \(t\) and its width is \(w\), then \(A = wt\).
$$I = (wt)nvq$$ $$v = \frac{I}{ntwq}$$
Step 3: Calculate \(V_H\)
Substitute \(v\) back into the electric field equation \(E = Bv\), and then use \(V_H = Ew\):
$$V_H = (Bv) w$$ $$V_H = B \left( \frac{I}{ntwq} \right) w$$
The width \(w\) cancels out, leaving the Hall voltage equation:
$$V_H = \frac{BI}{ntq}$$
Where:
- \(V_H\): Hall Voltage (V)
- \(B\): Magnetic Flux Density (T)
- \(I\): Current in the strip (A)
- \(n\): Number density of charge carriers (m\({^{-3}}\))
- \(t\): Thickness of the strip, perpendicular to \(B\) (m)
- \(q\): Charge of the carrier (C)
4.3 The Hall Probe
The Hall effect is extremely useful because it allows us to measure magnetic field strength (\(B\)).
- A Hall probe is simply a semiconductor strip designed to maximize \(V_H\).
- Since \(V_H \propto B\) (if \(I, n, t, q\) are fixed), the Hall voltage reading is directly proportional to the magnetic flux density.
- Semiconductors are used because they have a much lower number density (\(n\)) compared to metals, which leads to a much larger and more easily measurable Hall voltage.
Students often forget that the thickness \(t\) in the Hall voltage formula is the dimension of the strip parallel to the magnetic field. Make sure you correctly identify which dimension is \(t\) in a given problem.
5. Velocity Selection
5.1 The Principle of Velocity Selection
A velocity selector is a clever device that uses crossed (perpendicular) electric (\(E\)) and magnetic (\(B\)) fields to isolate particles moving at a specific speed (\(v\)).
How it works:
Imagine firing a beam of charged particles into a region where:
- The Electric Field (\(E\)) pulls the charges *upwards* (Force \(F_E\)).
- The Magnetic Field (\(B\)) is directed such that the magnetic force (\(F_B\)) pulls the charges *downwards* (or vice versa, depending on the charge).
If the forces balance exactly, the particle will experience zero net force and travel straight through without deflection.
5.2 Deriving the Selected Velocity
For the forces to be balanced:
$$F_{electric} = F_{magnetic}$$
Using the definitions for electric force (\(F_E = EQ\)) and magnetic force (\(F_B = BQv\)), and assuming \(\theta = 90^\circ\):
$$EQ = BQv$$
The charge \(Q\) cancels out (this is fantastic—the selected velocity is the same regardless of the charge magnitude or sign!)
$$E = Bv$$
Rearranging to find the selected velocity \(v\):
$$v = \frac{E}{B}$$
Only particles moving at this exact speed \(v\) will pass straight through the selector. Particles moving slower will be deflected by the stronger electric force, and particles moving faster will be deflected by the stronger magnetic force.
Did you know? Velocity selection is a fundamental component of mass spectrometry, allowing scientists to ensure all particles entering the main analysis chamber have the same kinetic energy before being separated by mass.
Chapter Key Takeaways
1. Force Magnitude: The force on a moving charge is calculated by \(F = BQv \sin \theta\).
2. Force Direction: Use Fleming’s LHR (Middle Finger = \(+Q\) motion, Forefinger = \(B\), Thumb = \(F\)).
3. Circular Motion: If \(v\) is perpendicular to \(B\), the force provides centripetal acceleration, resulting in a radius \(r = \frac{mv}{BQ}\).
4. Hall Effect: A transverse voltage \(V_H\) develops across a current-carrying conductor in a magnetic field, proportional to \(B\). The governing equation is \(V_H = \frac{BI}{ntq}\).
5. Velocity Selector: By balancing electric and magnetic forces, we can select particles moving at a specific speed \(v = E/B\).