Magnetic Fields

AS UNIT 4

Fields and Options

4.4 Magnetic Fields

Learners should be able to demonstrate and apply their knowledge and understanding of:
a) How to determine the direction of the force on a current carrying conductor in a magnetic field
b) How to calculate the magnetic field, B, by considering the force on a current carrying conductor in a magnetic field i.e. understand how to use F = BIL sin⁡θ
c) How to use F = Bqv sinθ for a moving charge in a magnetic field
d) The processes involved in the production of a Hall voltage and understand that [math]V_H ∝ B[/math] for constant I
e) The shapes of the magnetic fields due to a current in a long straight wire and a long solenoid
f) The equations [math] B = \frac{μ_0 I}{2πα} \, \text{and} B = μ_0 nI[/math]  for the field strengths due to a long straight wire and in a long solenoid
g) The fact that adding an iron core increases the field strength in a solenoid
h) The idea that current carrying conductors exert a force on each other and to predict the directions of the forces
i) Quantitatively, how ion beams of charged particles, are deflected in uniform electric and magnetic fields
j) The motion of charged particles in magnetic and electric fields in linear accelerators, cyclotrons and synchrotrons

Specified Practical Work

o   Investigation of the force on a current in a magnetic field

o   Investigation of magnetic flux density using a Hall probe

  • a) Determining the Direction of the Force on a Current-Carrying Conductor in a Magnetic Field

  • A current-carrying conductor in a magnetic field experiences a force due to the interaction between the moving charges in the conductor and the external magnetic field. The direction of this force is given by the Fleming’s Left-Hand Rule:
  • ⇒ Fleming’s Left-Hand Rule:
  • 1. Thumb → Direction of the force (F)
  • 2. First Finger → Direction of the magnetic field (B) (from North to South)
  • 3. Middle Finger → Direction of the current (I) (from positive to negative)
  • Figure 1 Left- Hand Rule
  • ⇒ Example:
  • If a wire carries a current upwards and the magnetic field points from left to right, then the force will be directed out of the page (towards you).

  • b) Using the Equation [math]F = BIL sinθ[/math]

  • This equation calculates the force on a straight current-carrying conductor in a uniform magnetic field:
  • [math]F = BIL sinθ[/math]
  • Where:
  • – F = force on the conductor (N)
  • – B = magnetic field strength (T, Tesla)
  • – I = current in the conductor (A)
  • – L = length of the conductor in the magnetic field (m)
  • – θ = angle between the conductor and the magnetic field
  • ⇒ Special Cases:
  • – If [math]θ = 0^0[/math] (conductor is parallel to the field), then [math]0 = 0^0[/math] → No force.
  • – If [math]θ = 90^0[/math] (conductor is perpendicular to the field), then [math]90^o = 1[/math] → Maximum force.
  • [math]B = \frac{F}{I L} \quad \text{at} \quad \theta = 90^\circ[/math]
  • Since magnetic fields exert a force on moving charges, we can drive a formula for this by considering a charge q moving through a magnetic field B. If the charge is moving at a  constant velocity v it is creating a current ( since it is a moving charge ), and this current moves a distance l in a time t, So
  • [math]v = \frac{l}{t} \\ l = v t[/math]
  • ⇒ Example Calculation
  • A 2 m long wire carries 5 A of current and is placed perpendicular to a 0.3 T magnetic field.
  • [math]F = BIL \sin\theta \\
    F = (0.3)(5)(2) \sin 90^\circ \\
    F = 3N[/math]
  • So, the force on the wire is 3 N.

  • c) Using the Equation [math]F = Bqv sin θ[/math]  for a Moving Charge in a Magnetic Field

  • When a charged particle moves through a magnetic field, it experiences a Lorentz force, given by:
  • [math]F = Bqv sin θ[/math]
  • Where:
  • – F = force on the charge (N)
  • – B = magnetic field strength (T)
  • – q = charge of the particle (C)
  • – v = velocity of the particle (m/s)
  • – θ = angle between velocity and magnetic field
  • ⇒ Effects
  • – If [math]θ = 0^0[/math] or [math]180^0[/math], then [math]sin 0^0 = 0[/math] → No force (particle moves unaffected).
  • – If [math]θ = 90^0[/math], then [math]sin 90^0 = 1[/math] → Maximum force, causing circular motion.
  • ⇒ Example Calculation
  • A proton ( [math]q = 1.6 × 10^{-19} C[/math]) moves at [math]2 × 10^6 m/s[/math] perpendicular to a 0.1 T field.
  • [math]F = Bqv \sin\theta \\
    F = (0.1)(1.6 \times 10^{-19})(2 \times 10^6) \sin 90^\circ \\
    F = 3.2 \times 10^{-14} \, N[/math]
  • The force on the proton is [math]3.2 × 10^{-14} N[/math].

  • d) Hall Voltage Production and [math]V_H ∝ B[/math]  for Constant I

  • 1. The Hall Effect and Hall Voltage
  • The Hall Effect occurs when a current-carrying conductor or semiconductor is placed perpendicular to a magnetic field. The moving charge carriers (electrons in a conductor or holes in a p-type semiconductor) experience a Lorentz force due to the external magnetic field.
  • This force causes the charge carriers to accumulate on one side of the material, creating a potential difference known as the Hall voltage ([math]V_H [/math] ​).
  • Figure 2 Hall Effect
  • 2. Derivation of Hall Voltage
  • The force experienced by a charge moving in a magnetic field is:
  • [math]F = Bqv[/math]
  • Where:
  • – B = magnetic field strength (T)
  • – q = charge of carrier (C)
  • – v = drift velocity of carriers (m/s)
  • At equilibrium, this force is balanced by the electric force due to the Hall voltage:
  • [math]F = qE[/math]
  • Since[math]E = V_H/d[/math] , where d is the thickness of the conductor, we get:
  • [math]Bqv = q \frac{V_H}{d}[/math]
  • Simplifying:
  • [math]V_H = Bvd[/math]
  • Since current (I) is related to charge carrier density n:
  • [math]I = nqvA[/math]
  • Where A is the cross-sectional area of the conductor, we substitute [math]v = I/(nqA)[/math]:
  • [math]V_H = \frac{B I}{n q A d}[/math]
  • Since for a given material n, q, and A are constant, we get:
  • [math]V_H = B[/math]
  • This means Hall voltage is directly proportional to the magnetic field strength when current I is kept constant.
  • ⇒ Applications of the Hall Effect:
  • – Measuring magnetic field strength in Hall sensors
  • – Determining whether a semiconductor is n-type or p-type
  • – Measuring current in high-power circuits

  • e) The shapes of the magnetic fields due to a current in a long straight wire and a long solenoid

  • ⇒ Shape of the Field
  • – A long straight current-carrying wire generates a circular magnetic field around it.
  • – The field lines form concentric circles centered on the wire.
  • – The strength of the field decreases as you move farther from the wire.
  • ⇒ Direction of the Field
  • The direction of the field is determined using the Right-Hand Rule:
  • – Thumb points in the direction of the current (I).
  • – Curled fingers show the direction of the magnetic field (B).
  • Figure 3 Direction of current and magnetic field
  • ⇒ Example:
  • If the current is flowing upward, the magnetic field circles counterclockwise when viewed from above.

  • f) Magnetic Fields Due to a Current-Carrying Wire and Solenoid

  • 1. Magnetic Field Around a Long Straight Wire
  • A long straight current-carrying wire produces a circular magnetic field around it, as shown by concentric field lines.
  • The direction of the field is given by the Right-Hand Rule:
  • – Thumb → Direction of current (I)
  • – Fingers → Direction of magnetic field lines
  • The magnetic field strength at a distance α from the wire is:
  • [math]B = \frac{\mu_0 I}{2\pi \alpha}[/math]
  • Where:
  • – B = magnetic field strength (T)
  • – [math]\mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A}[/math] (Permeability of free space)
  • – I = current in the wire (A)
  • – α = perpendicular distance from the wire (m)
  • ⇒ Observations:
  • – [math]B ∝ l[/math] (stronger current = stronger field)
  • – ​[math]B ∝ \frac{1}{α}[/math] (closer to the wire = stronger field)
  • 2. Magnetic Field Inside a Long Solenoid
  • A solenoid is a coil of wire that generates a uniform magnetic field inside it when current flows.
  • The field inside a long solenoid is given by:
  • [math]B = μ_0 nI[/math]
  • Where:
  • – [math]n = \frac{N}{L}[/math]​ (number of turns per unit length, in turns/m)
  • – B = magnetic field strength (T)
  • – I = current in the solenoid (A)
  • Figure 4 Magnetic Field inside a solenoid
  • ⇒ Observations:
  • – Inside the solenoid: Field lines are parallel and evenly spaced, meaning a strong uniform field.
  • – Outside the solenoid: The field is weak and spreads out, similar to a bar magnet.
  • ⇒ Applications of Solenoids:
  • – Electromagnets (in motors, relays, and MRI machines)
  • – Inductors in electrical circuits
  • – Magnetic field shielding

  • g) How an Iron Core Increases the Magnetic Field Strength in a Solenoid

  • A solenoid is a coil of wire that generates a magnetic field when current flows through it. The strength of this magnetic field depends on:
  • 1. The number of turns per unit length (n)
  • 2. The current flowing through the wire (I)
  • 3. The presence of a magnetic core, such as iron
  • Figure 5 Magnetic field strength in an AC Coil
  • ⇒ Effect of an Iron Core
  • If a solenoid is wrapped around an iron core, the magnetic field inside the solenoid increases significantly.
  • This happens because iron is highly permeable, meaning it easily allows magnetic field lines to pass through it and enhances the magnetic flux.
  • The new field strength inside the solenoid is given by:
  • [math]B = μnI[/math]
  • Where:
  • – B = magnetic field strength (T)
  • – [math]μ = μ_0 μ_r[/math] is the permeability of the core material
  • – [math]μ_r[/math] = relative permeability of the material (for iron, ​[math]μ_r ≈ 5000[/math] , meaning it greatly amplifies the field)
  • – n = turns per unit length (turns/m)
  • – I = current in the solenoid (A)
  • ⇒ Practical Applications of Adding an Iron Core
  • – Electromagnets in electric motors, transformers, and magnetic relays
  • – MRI machines to generate strong, controlled magnetic fields
  • – Inductors in electronic circuits

  • h) Force Between Two Current-Carrying Conductors

  • Two parallel current-carrying wires exert forces on each other due to the magnetic fields they generate.
  • ⇒ Predict the Direction of Forces
  • If currents flow in the same direction, the wires attract each other.
  • If currents flow in opposite directions, the wires repel each other.
  • Figure 6 Force between two current carrying conductor
  • – This can be explained using Ampère’s Law and the Right-Hand Rule.
  • ⇒ Quantitative Expression of Force
  • The force per unit length between two parallel wires separated by distance d is given by:
  • [math]\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}[/math]
  • Where:
  • – F = force (N)
  • – L = length of the wires (m)
  • – [math]l_1, l_2[/math]​ ​ = currents in the wires (A)
  • – d = distance between the wires (m)
  • – ​[math]μ_0[/math] = permeability of free space ([math]4π × 10^{-7}Tm/A[/math] )
  • ⇒ Applications of Current-Carrying Conductors Exerting Forces
  • Loudspeakers (force on a coil moves the speaker diaphragm)
  • Magnetic levitation (MagLev) trains
  • Fusion reactors, where strong magnetic fields confine plasma

  • i) Deflection of Ion Beams in Electric and Magnetic Fields

  • Charged particles, such as ions or electrons, experience forces when moving through electric or magnetic fields.
  • 1. Motion in a Uniform Electric Field
  • A charged particle in an electric field experiences a constant force:
  • [math]F = qE[/math]
  • Where:
  • – F = force on the particle (N)
  • – q = charge of the particle (C)
  • – E = electric field strength (V/m)
  • Figure 7 Uniform electric field
  • If an ion beam moves perpendicular to an electric field, it undergoes uniform acceleration, similar to projectile motion.
  • The deflection follows a parabolic
  • ⇒ Applications:
  • Cathode ray tubes (old TVs, oscilloscopes)
  • 2. Motion in a Uniform Magnetic Field
  • A charged particle moving in a magnetic field perpendicular to its velocity experiences a force:
  • [math]F = qE[/math]
  • This force acts perpendicular to the motion, causing the particle to move in a circular path with radius:
  • [math]r = \frac{mv}{Bq}[/math]
  • Where:
  • – m = mass of the particle (kg)
  • – v = velocity of the particle (m/s)
  • – q = charge of the particle (C)
  • – B = magnetic field strength (T)
  • Figure 8 Uniform Magnetic Field
  • ⇒ Applications: Particle accelerators, mass spectrometry

  • j) Charged Particle Motion in Accelerators

  • Particle accelerators use electric and magnetic fields to control charged particles.
  • 1. Linear Accelerator (Linac)
  • o   Uses a series of electric fields to accelerate charged particles in a straight line.
  • o   Used in medical treatments (radiotherapy) and research.
  • 2. Cyclotron
  • Uses a magnetic field to keep particles in a spiral path while an alternating electric field accelerates them.
  • The equation for the cyclotron frequency is:
  • [math]f = \frac{qB}{2\pi m}[/math]
  • ⇒ Applications: Proton therapy, nuclear physics
  • Figure 9 Cyclotron Principle
  • 3. Synchrotron
  • Similar to a cyclotron, but adjusts the magnetic field as the particle gains speed to keep it on the same path.
  • Used in high-energy physics, like the Large Hadron Collider (LHC).
  • Figure 10 Synchrotron

  • Specified Practical Work

  • 1. Investigation of the Force on a Current in a Magnetic Field

  • This experiment aims to study the force exerted on a current-carrying conductor placed in a magnetic field and verify the relationship:
  • [math]F = BILsinθ[/math]
  • Where:
  • – F = Force on the conductor (N)
  • – B = Magnetic flux density (T)
  • – I = Current in the conductor (A)
  • – L = Length of conductor in the field (m)
  • – θ = Angle between conductor and field
  • Figure 11 Force on a current carrying conductor
  • Apparatus
  • – Magnet (U-shaped or a pair of strong permanent magnets)
  • – Rectangular conducting wire (placed between the magnetic poles)
  • – Variable DC power supply (to control current)
  • – Ammeter (to measure current)
  • – Electronic balance (to measure force)
  • – Ruler (to measure length of conductor)
  • ⇒ Procedure
  • 1. Setup the Circuit
  • – Place a straight conducting wire perpendicular to a uniform magnetic field (between two magnets).
  • – Connect the wire to a variable DC power supply and ammeter in series.
  • 2. Measure Initial Weight
  • – Place the entire setup (including wire and support) on an electronic balance.
  • – Record the initial mass reading ( ​[math]m_o[/math]).
  • 3. Pass a Current Through the Wire
  • – Adjust the power supply to pass a small known current (I) through the wire.
  • – The force exerted on the wire by the magnetic field will cause a measurable change in the balance reading.
  • 4. Record the New Mass Reading
  • – Note the new mass (mmm) reading on the balance.
  • – The difference in mass ([math]Δm[/math] ) corresponds to the magnetic force:
  • [math]F = Δm ⋅ g[/math]
  • Where g = acceleration due to gravity (9.81 m/s²).
  • 5. Repeat for Different Currents
  • – Increase the current in steps and record the force each time.
  • 6. Analyze the Data
  • – Plot a graph of force F vs current I.
  • – The slope of the graph should be [math]BLsinθ[/math] allowing the calculation of B.
  • ⇒ Expected Results
  • The graph should be a straight line showing that force is directly proportional to current.
  • The slope of the graph can be used to calculate the magnetic field strength B.
  • ⇒ Sources of Error & Improvements
Error Improvement
Contact resistance in the circuit affects current measurements Use thick, low-resistance wires
Balance readings fluctuate due to vibrations Use damping methods to stabilize
Magnetic field may not be perfectly uniform Ensure magnet poles are aligned properly
Wire may not be perfectly perpendicular to the field Use clamps & alignment guides

  •  2. Investigation of Magnetic Flux Density Using a Hall Probe

  • A Hall probe is used to measure the magnetic flux density BBB of a magnetic field based on the Hall effect.
  • ⇒ Principle of the Hall Effect
  • When a current-carrying conductor is placed in a magnetic field, the moving charge carriers (electrons) experience a force (Lorentz force).
  • This causes the charges to accumulate on one side, creating a Hall voltage:
  • [math]V_H = \frac{B I}{n q t}[/math]
  • Where:
  • – [math]V_H[/math]​ = Hall voltage (V)
  • – B = Magnetic flux density (T)
  • – I = Current in the probe (A)
  • – n = Charge carrier density ([math]m^{-3}[/math])
  • – q = Charge of an electron ([math]1.6 × 10^{-19} C)[/math] )
  • – t = Thickness of the probe (m)
  • Figure 12 Magnetic flux density using Hall Probe
  • ⇒ Apparatus
  • – Hall probe
  • – Digital voltmeter (to measure Hall voltage)
  • – Electromagnet or permanent magnet
  • – Variable power supply
  • – Ammeter (to measure current in Hall probe)
  • ⇒ Procedure
  • 1. Set Up the Hall Probe
  • – Place the Hall probe in the region where B needs to be measured.
  • – Connect the probe to a power supply to allow current to flow through it.
  • 2. Align the Hall Probe in the Field
  • – Ensure the flat face of the probe is perpendicular to the magnetic field.
  • 3. Measure the Hall Voltage
  • – Read the Hall voltage ([math]V_H[/math] ​) from the digital voltmeter.
  • 4. Calculate Magnetic Flux Density BBB
  • – Use the equation:
  • [math]B = \frac{V_H n q t}{I}[/math]
  • 5. Repeat for Different Field Strengths
  • – If using an electromagnet, increase the magnetic field strength and repeat the measurements.
  • ⇒ Expected Results
  • The Hall voltage should be directly proportional to the magnetic flux density B.
  • Plotting ​[math]V_H[/math] vs B should yield a linear graph.
  • ⇒ Sources of Error & Improvements
Error Improvement
Hall probe may not be perfectly perpendicular Use mounting guides for alignment
Temperature changes affect readings Allow probe to reach thermal stability before measuring
Contact resistance in the circuit affects current Use low-resistance connections
Fluctuations in power supply voltage Use a regulated power supply
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