Chapter: Magnetic Fields
Welcome to your study notes for Magnetic Fields! Magnetism might seem like invisible magic at first, but by the end of this chapter, you will understand the exact rules that govern how magnetic fields interact with electric currents and moving charges. These concepts form the backbone of modern technology, powering everything from electric motors and particle accelerators to hospital MRI scanners and renewable energy generators.
Don't worry if some of the 3D spatial rules feel a bit unusual at first. We will break down every single equation, rule, and application step-by-step with clear memory aids to help you succeed in your CCEA A2 Physics examinations.
---1. Understanding Magnetic Fields
A magnetic field is a region of space where a magnetic pole, a current-carrying wire, or a moving charged particle experiences a magnetic force. Just like gravitational and electric fields, magnetic fields are vector fields, meaning they have both a magnitude (strength) and a direction.
Magnetic Field Lines
We represent magnetic fields visually using magnetic field lines (or lines of magnetic flux). Here are the key rules to remember:
• Field lines always point from the North pole to the South pole outside of a magnet.
• The closeness (density) of the field lines shows the strength of the field: closer lines mean a stronger magnetic field.
• Field lines never cross each other because the magnetic field can only have one resultant direction at any given point.
Field Patterns to Know
• Uniform Magnetic Field: Represented by straight, parallel, and equally spaced field lines (for example, between two wide, flat opposite magnetic poles).
• Long Straight Current-Carrying Wire: Creates concentric circular field lines around the wire. You can find their direction using the Right-Hand Grip Rule: point your right thumb in the direction of conventional current (\(+\) to \(-\)), and your curled fingers show the direction of the magnetic field.
• Solenoid (Coil of Wire): The field pattern outside looks identical to that of a bar magnet. Inside the solenoid, the field lines are straight, parallel, and closely spaced, producing a strong and nearly uniform magnetic field.
Quick Review: Magnetic field lines go North \(\rightarrow\) South. Closer lines \(\rightarrow\) stronger field.
---2. Force on a Current-Carrying Conductor
When an electric current flows through a wire placed inside an external magnetic field, the magnetic field produced by the moving electrons interacts with the external magnetic field. This creates a physical mechanical force on the wire. This phenomenon is known as the motor effect.
The Force Equation
The magnitude of the magnetic force \(F\) acting on a straight conductor of length \(L\) carrying a current \(I\) in a uniform magnetic field of flux density \(B\) is given by:
\(F = B I L \sin\theta\)
Where:
• \(F\) = Magnetic force acting on the wire, measured in newtons (\(\text{N}\))
• \(B\) = Magnetic flux density (the strength of the field), measured in tesla (\(\text{T}\))
• \(I\) = Current flowing through the wire, measured in amperes (\(\text{A}\))
• \(L\) = Length of the wire that is inside the magnetic field, measured in metres (\(\text{m}\))
• \(\theta\) = The angle between the direction of the magnetic field and the direction of the current
Special Cases for the Angle (\(\theta\))
• Perpendicular (\(\theta = 90^\circ\)): Since \(\sin(90^\circ) = 1\), the force is at its maximum: \(F = B I L\).
• Parallel (\(\theta = 0^\circ\) or \(180^\circ\)): Since \(\sin(0^\circ) = 0\), the magnetic force is zero (\(F = 0\)). A wire parallel to the field lines experiences no force at all!
Defining the Tesla (\(\text{T}\))
In your CCEA exam, you may be asked to define the unit of magnetic flux density, the tesla (\(\text{T}\)). Rearranging the equation for perpendicular alignment gives \(B = \frac{F}{I L}\):
One Tesla is defined as the magnetic flux density that produces a force of \(1\text{ N}\) per metre on a conductor carrying a current of \(1\text{ A}\) perpendicular to the magnetic field.
Unit equivalence: \(1\text{ T} = 1\text{ N}\cdot\text{A}^{-1}\cdot\text{m}^{-1}\)
Finding the Direction: Fleming's Left-Hand Rule
To find the direction of the resulting force, hold your left hand with your thumb, first finger, and second finger all mutually at right angles (\(90^\circ\) to each other):
• First finger = Magnetic Field (pointing from North to South)
• Second finger = Current (conventional current, positive to negative)
• Thumb = Thrust / Motion (the direction of the magnetic Force)
Memory Trick: Think of the letters FBI:
• Force (Thumb)
• B-field (First Finger)
• I-current (Second Finger)
Common Mistake to Avoid: Always remember that the second finger represents conventional current (from positive to negative). If an exam question mentions the flow of electrons, remember that electrons flow in the opposite direction to conventional current!
Key Takeaway: \(F = B I L \sin\theta\). The force is maximum when the conductor is perpendicular to the field and zero when parallel. Use your left hand (FBI) to find the direction.
---3. Force on a Moving Charged Particle
Since an electric current is simply a flow of moving charges, an individual charged particle moving through a magnetic field also experiences a magnetic force.
Deriving the Equation
Consider a particle with charge \(q\) moving at velocity \(v\) through a field of flux density \(B\). Over a time \(\Delta t\), it travels a distance \(L = v \Delta t\). The current created by this charge is \(I = \frac{q}{\Delta t}\).
Substituting these into \(F = B I L \sin\theta\):
\(F = B \left(\frac{q}{\Delta t}\right) (v \Delta t) \sin\theta = B q v \sin\theta\)
Where:
• \(F\) = Magnetic force on the charged particle (\(\text{N}\))
• \(B\) = Magnetic flux density (\(\text{T}\))
• \(q\) = Magnitude of the charge on the particle (\(\text{C}\))
• \(v\) = Velocity of the particle (\(\text{m}\cdot\text{s}^{-1}\))
• \(\theta\) = Angle between the particle's velocity vector and the magnetic field lines
Important Direction Rules for Charged Particles
• Positive charges (e.g. protons, \(\alpha\)-particles): The direction of motion is the same as the conventional current. Point your second finger in the direction of velocity \(v\).
• Negative charges (e.g. electrons, \(\beta\)-particles): The conventional current points in the opposite direction to their velocity. Point your second finger opposite to the direction the electron is moving!
4. Circular Motion of Charged Particles in a Magnetic Field
What happens when a charged particle enters a uniform magnetic field at right angles (\(\theta = 90^\circ\))?
According to Fleming's Left-Hand Rule, the magnetic force \(F\) is always perpendicular to both the magnetic field \(B\) and the velocity \(v\). Because the force is constantly perpendicular to the velocity:
1. No work is done on the particle by the magnetic force (since work done \(W = F s \cos(90^\circ) = 0\)).
2. The particle's speed and kinetic energy remain constant.
3. Only the direction of motion changes continuously.
4. The magnetic force acts as a centripetal force, causing the particle to move along a circular path!
Deriving the Orbital Radius
Equating the magnetic force to the centripetal force formula:
\(F_{\text{magnetic}} = F_{\text{centripetal}}\)
\(B q v = \frac{m v^2}{r}\)
Cancelling one \(v\) on both sides and solving for the radius \(r\):
\(r = \frac{m v}{B q}\)
Where:
• \(r\) = Radius of the circular path (\(\text{m}\))
• \(m\) = Mass of the particle (\(\text{kg}\))
• \(v\) = Speed of the particle (\(\text{m}\cdot\text{s}^{-1}\))
• \(B\) = Magnetic flux density (\(\text{T}\))
• \(q\) = Charge of the particle (\(\text{C}\))
Key Insights from the Radius Formula:
• Larger momentum (\(p = mv\)) \(\implies\) larger radius (wider curve).
• Stronger magnetic field (\(B\)) \(\implies\) smaller radius (tighter curve).
• Larger charge (\(q\)) \(\implies\) smaller radius (tighter curve).
Period and Frequency of the Orbit
The time \(T\) taken for the particle to complete one full circular orbit of circumference \(2\pi r\) is:
\(T = \frac{2\pi r}{v}\)
Substituting \(r = \frac{m v}{B q}\) into this expression:
\(T = \frac{2\pi \left(\frac{m v}{B q}\right)}{v} = \frac{2\pi m}{B q}\)
Crucial CCEA Exam Point: Notice that the speed \(v\) cancels out! The time period \(T\) (and therefore frequency \(f = \frac{1}{T} = \frac{B q}{2\pi m}\)) is independent of the particle's speed and orbital radius. Faster particles travel around a larger circle, taking exactly the same amount of time per orbit as slower particles on smaller circles!
---5. Applications of Magnetic and Electric Fields
Application 1: The Velocity Selector
A velocity selector is a device that allows charged particles of only one specific speed to pass through undeflected, regardless of their mass or charge.
It consists of perpendicular uniform electric (\(E\)) and magnetic (\(B\)) fields (crossed fields):
• The electric field exerts a force: \(F_E = q E\)
• The magnetic field exerts an opposing force: \(F_B = B q v\)
For a particle to travel in a perfectly straight line without being deflected, the two opposing forces must be equal in magnitude:
\(q E = B q v \implies v = \frac{E}{B}\)
• Particles with speed \(v = \frac{E}{B}\) emerge in a straight line through the exit slit.
• Particles with \(v > \frac{E}{B}\) experience a stronger magnetic force and curve towards the magnetic force side.
• Particles with \(v < \frac{E}{B}\) experience a stronger electric force and curve towards the electric force side.
Application 2: The Mass Spectrometer
A mass spectrometer separates ions based on their mass-to-charge ratio (\(\frac{m}{q}\)).
1. Ions are first selected to have a uniform speed \(v\) using a velocity selector.
2. They enter a region containing only a uniform magnetic field \(B\), where they follow semicircular paths of radius \(r = \frac{m v}{B q}\).
3. By rearranging the radius formula: \(\frac{m}{q} = \frac{B r}{v}\). Measuring the radius of curvature allows scientists to accurately determine the mass of unknown isotopes and molecules.
Application 3: The Cyclotron
A cyclotron is a circular particle accelerator used to produce high-energy beams of protons or ions for nuclear physics research and medical treatments (such as cancer radiotherapy and creating radioactive medical tracers).
Structure:
• Two hollow, D-shaped metal chambers called Dees placed inside a vacuum chamber.
• A uniform magnetic field acts perpendicularly through the flat faces of the Dees.
• A high-frequency alternating potential difference is applied across the narrow gap between the Dees.
How it works step-by-step:
1. An ion source at the centre releases charged particles.
2. The magnetic field causes the particles to move in a semicircular path inside the Dee.
3. Inside the metal Dees, the electric field is zero (due to electrostatic shielding), so the particles travel at constant speed.
4. When the particle reaches the gap between the Dees, the electric field accelerates it across the gap, increasing its kinetic energy.
5. The particle enters the opposite Dee with greater speed. Because \(r = \frac{m v}{B q}\), it follows a semicircle of larger radius.
6. Since the time spent in each Dee is constant (\(t = \frac{T}{2} = \frac{\pi m}{B q}\)), the alternating voltage is timed at a fixed frequency to reverse its polarity every time the particle reaches the gap.
7. The particle spirals outward, gaining energy with each gap crossing, until it reaches the outer edge and is deflected toward a target.
Key Takeaway: In a cyclotron, the magnetic field controls the direction (causes circular motion), while the electric field provides the acceleration (increases kinetic energy) across the gap.
---6. Electromagnetic Induction
Whenever there is relative movement between a conductor and a magnetic field, or whenever the magnetic field passing through a coil changes, an electromotive force (e.m.f.) is induced. This process is called electromagnetic induction.
Magnetic Flux (\(\Phi\))
Magnetic flux is a measure of the total amount of magnetic field passing through a given area \(A\). Think of it like catching rain in a bucket: the amount of water caught depends on the rain intensity and the orientation of the bucket.
\(\Phi = B A \cos\theta\)
Where:
• \(\Phi\) = Magnetic flux, measured in webers (\(\text{Wb}\))
• \(B\) = Magnetic flux density (\(\text{T}\))
• \(A\) = Cross-sectional area (\(\text{m}^2\))
• \(\theta\) = Angle between the magnetic field lines and the normal (perpendicular line) to the surface of the coil
• When the magnetic field is perpendicular to the plane of the coil (\(\theta = 0^\circ\)): \(\Phi = B A\).
• When the magnetic field is parallel to the plane of the coil (\(\theta = 90^\circ\)): \(\Phi = 0\).
• Unit relationship: \(1\text{ Wb} = 1\text{ T}\cdot\text{m}^2\)
Magnetic Flux Linkage (\(N\Phi\))
If a coil has \(N\) turns of wire, each turn passes through the magnetic flux. The total flux passing through the entire coil is called the magnetic flux linkage:
\(\text{Magnetic Flux Linkage} = N \Phi = B A N \cos\theta\)
Units: weber-turns (\(\text{Wb}\)) or simply \(\text{Wb}\).
Faraday's Law of Electromagnetic Induction
The magnitude of the induced electromotive force (e.m.f.) is directly proportional to the rate of change of magnetic flux linkage.
Lenz's Law
The direction of the induced e.m.f. (and induced current) is always such that it opposes the change in magnetic flux that produces it.
Lenz's Law is a direct consequence of the Conservation of Energy. If the induced current assisted the change rather than opposing it, energy would be created from nothing, violating fundamental physical laws!
Combined Mathematical Formula
Combining Faraday's Law and Lenz's Law gives the fundamental equation of induction:
\(\varepsilon = -\frac{\Delta (N\Phi)}{\Delta t}\)
Where:
• \(\varepsilon\) = Induced electromotive force (\(\text{V}\))
• \(\frac{\Delta (N\Phi)}{\Delta t}\) = Rate of change of magnetic flux linkage (\(\text{Wb}\cdot\text{s}^{-1}\))
• The minus sign (\(-\)) represents Lenz's Law (opposition to the change).
Induced e.m.f. in a Straight Conductor Moving Through a Field
When a straight rod of length \(L\) moves at a constant speed \(v\) perpendicular to a uniform magnetic field \(B\), the induced e.m.f. across its ends is:
\(\varepsilon = B L v\)
---7. Summary & Quick Revision Checklist
Before sitting your CCEA exam, make sure you can comfortably do the following:
• State the definition and units of magnetic flux density \(B\) (\(\text{T}\)), magnetic flux \(\Phi\) (\(\text{Wb}\)), and flux linkage \(N\Phi\) (\(\text{Wb-turns}\)).
• Use \(F = B I L \sin\theta\) and apply Fleming's Left-Hand Rule to find the direction of force on a current-carrying wire.
• Use \(F = B q v \sin\theta\) for moving charges, remembering that negative charges move in the opposite direction to conventional current.
• Explain why charged particles travel in a circle in a uniform magnetic field and derive \(r = \frac{m v}{B q}\).
• Explain the role of electric and magnetic fields in a velocity selector (\(v = \frac{E}{B}\)) and in a cyclotron.
• State Faraday's Law and Lenz's Law and use \(\varepsilon = -\frac{\Delta (N\Phi)}{\Delta t}\) to solve numerical problems.