represent a magnetic field by field lines

Representing a Magnetic Field by Field Lines – Cambridge IGCSE/A‑Level (9702)

Learning Outcomes

  • Define the magnetic field vector B and magnetic flux density (tesla, T).

  • Draw and interpret magnetic‑field‑line patterns for the main sources required by the syllabus.

  • Apply the right‑hand grip rule (current‑carrying conductors) and Fleming’s left‑hand rule (force on a current‑carrying conductor).

  • Calculate magnetic forces on moving charges and on current‑carrying conductors (AO2).

  • Explain magnetic flux, Faraday’s law, Lenz’s law and induced e.m.f. (AO1 / AO2).

  • Describe experimental techniques for visualising magnetic fields and for measuring B (AO3).

  • Link magnetic‑field concepts to alternating currents, inductance, quantum magnetic moments and medical imaging (MRI) (AO1 / AO2).

1. What Is a Magnetic Field?

A magnetic field is a region of space in which a moving charge or a current‑carrying conductor experiences a magnetic force. The magnetic flux density B (units T) quantifies the strength and direction of this field.

For a charge q moving with velocity v:

\[\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}\tag{1}\]

For a straight conductor of length L carrying current I in a uniform field B:

\[\mathbf{F}=I\,\mathbf{L}\times\mathbf{B}\tag{2}\]

The direction of B at any point is defined as the direction of the force that would act on a positive test charge moving perpendicular to the field.

2. Magnetic Field Lines – Definition and Rules

  • The tangent to a field line at any point gives the direction of B there.

  • Line density (how close the lines are) is proportional to the magnitude of B; closer lines → stronger field.

  • Field lines form continuous closed loops – they never start or end in empty space.

  • Outside a permanent magnet they run from the north (N) pole to the south (S) pole; inside the magnet they run from S to N.

  • Lines never intersect (a single point cannot have two different directions of B).

  • The total number of lines drawn is proportional to the strength of the source (current, magnetic moment, etc.).

3. Hand‑Rule Summaries

3.1 Right‑Hand Grip Rule (Current‑Carrying Conductor)

Thumb → direction of conventional current I.

Curling fingers → direction of the magnetic field lines that encircle the wire.

3.2 Fleming’s Left‑Hand Rule (Force on a Current‑Carrying Conductor)

First finger → magnetic field B (from N to S).

Second finger → current I (conventional).

Thumb → force F on the conductor.

4. Magnetic Field of the Main Sources

SourceField‑Line PatternExpression for BKey Features (AO2)
Long straight wireConcentric circles centred on the wire\[B=\frac{\mu_{0}I}{2\pi r}\]Field strength ∝ 1⁄r; direction given by the right‑hand grip rule.
Current loop (single turn)Closed loops emerging from the centre; inside the loop the field is approximately uniform.\[B{\text{axis}}=\frac{\mu{0} I R^{2}}{2(R^{2}+r^{2})^{3/2}}\]On the axis the direction follows the right‑hand rule; far from the loop B ∝ 1⁄r³.
Solenoid (long coil, n turns / m)Parallel lines inside, spreading out at the ends\[B=\mu_{0} n I\quad\text{(ideal, inside)}Field inside is nearly uniform; external field is weak.
Toroid (solenoid bent into a ring)Concentric circles inside the core; essentially no external field\[B=\frac{\mu_{0} N I}{2\pi r}\quad(r=\text{mean radius})\]Useful for magnetic shielding; field confined to the core.
Bar magnet (magnetic dipole)Closed loops from N to S outside, S→N inside\[B=\frac{\mu_{0}}{4\pi}\frac{m}{r^{3}}\left(2\cos\theta\,\hat r+\sin\theta\,\hat\theta\right)\]Field strongest near the poles; inside the material the field is essentially uniform.
EarthLarge‑scale dipole; lines emerge near the geographic south pole and enter near the north pole.≈ 50 µT at the surface (varies with latitude)Provides a reference field for many laboratory experiments.

5. Magnetic Flux and Electromagnetic Induction

5.1 Magnetic Flux

Magnetic flux (Φ) through a surface of area A that makes an angle θ with the field is

\[\Phi = B A \cos\theta\qquad\text{(weber, Wb)}\]

Φ is a scalar; its sign indicates the sense of the field relative to the chosen normal direction.

5.2 Faraday’s Law

The magnitude of the induced e.m.f. (ε) in a closed loop is proportional to the rate of change of magnetic flux:

\[\lvert\varepsilon\rvert = \left|\frac{d\Phi}{dt}\right|\tag{3}\]

For a coil of N turns, ε = −N dΦ/dt. The negative sign is Lenz’s law (see below).

5.3 Lenz’s Law

The direction of the induced e.m.f. (and hence the induced current) is such that the magnetic field it creates opposes the change in the original flux.

5.4 Simple Induced‑e.m.f. Examples

  • Moving conductor in a uniform field: a rod of length ℓ moving with speed v perpendicular to B generates

    \[\varepsilon = B\ell v\]

    (useful for velocity‑selector problems).

  • Rotating coil (AC generator): for a coil of N turns, area A rotating at angular speed ω in a uniform field,

    \[\varepsilon = N B A \,\omega \sin(\omega t).\]

6. Forces Involving Magnetic Fields

6.1 Force on a Moving Charge

\[F = q v B \sin\theta\]

θ is the angle between v and B. Direction from the right‑hand rule for the cross product.

6.2 Force on a Current‑Carrying Conductor

\[F = I L B \sin\theta\]

Use Fleming’s left‑hand rule to determine the direction.

7. Alternating Currents, Inductance & Transformers (Link to AC)

  • A time‑varying current in a coil produces a time‑varying magnetic field, which according to Faraday’s law induces an e.m.f. in any nearby coil (mutual induction).

  • Self‑inductance L of a coil:

    \[\varepsilon_{\text{self}} = -L\frac{dI}{dt}\]

    where \(L = \frac{N\Phi}{I}\) (henry, H).

  • Transformer principle: a primary coil of N₁ turns creates a changing flux; a secondary coil of N₂ turns experiences an induced e.m.f.

    \[\frac{V{s}}{V{p}} = \frac{N{s}}{N{p}}\] (ideal transformer, neglecting losses).

8. Experimental Techniques for Visualising Magnetic Fields (AO3)

  • Iron‑filings method: Sprinkle fine iron filings on a sheet of paper placed over the source; the filings align with the local direction of B, revealing the line pattern.

  • Compass‑grid method: A grid of tiny magnetic compasses (or “magnetic needles”) shows the direction of the field at many points simultaneously.

  • Hall‑probe mapping: A calibrated Hall sensor is moved systematically across a region; the recorded voltage is converted to B, giving a quantitative field map.

  • Ferrofluid viewer: Ferrofluid droplets form spikes along field lines, useful for three‑dimensional visualisation.

  • Search‑coil (current‑probe) technique: A small coil linked to a galvanometer measures the induced emf proportional to the rate of change of magnetic flux, allowing indirect determination of B.

9. Extensions – Quantum & Medical Physics

9.1 Quantum Magnetic Moment

Elementary particles possess an intrinsic magnetic moment \(\boldsymbol{\mu}= \gamma \mathbf{S}\) where γ is the gyromagnetic ratio and S the spin angular momentum. The Stern‑Gerlach experiment demonstrated that magnetic moments are quantised, providing a bridge between magnetic‑field concepts and the quantum‑physics part of the syllabus.

9.2 Magnetic Resonance Imaging (MRI)

In MRI a strong, uniform static field (≈ 1–3 T) aligns the nuclear magnetic moments of hydrogen atoms in the body. Radio‑frequency pulses produce a changing magnetic field that, via Faraday’s law, induces detectable voltages in receiver coils. Understanding field uniformity, gradient fields and the relationship \( \varepsilon = -\frac{d\Phi}{dt}\) is essential for the medical‑physics component of the syllabus.

10. Worked Example (AO2)

Problem: A rectangular loop of width 0.10 m and height 0.20 m carries a current of 3 A clockwise (viewed from above). It is placed in a uniform magnetic field of 0.25 T directed into the page. Determine the magnitude and direction of the net force on the loop and comment on any torque.

  1. Identify the four sides: two vertical sides (L = 0.20 m) and two horizontal sides (L = 0.10 m).

  2. Force on each side using \(F = I L B \sin\theta\):

    • Vertical sides: \(\theta = 90^{\circ}\) → \(F_{\text{vert}} = 3 \times 0.20 \times 0.25 = 0.15\;\text{N}\) each.

    • Direction (Fleming’s left‑hand rule): left side upward, right side downward → forces cancel.

    • Horizontal sides: \(\theta = 0^{\circ}\) → \(F = 0\).

  3. Net force = 0 N.

  4. Torque: each vertical side produces a moment about the centre, \(\tau = F \times \frac{w}{2} = 0.15 \times 0.05 = 7.5\times10^{-3}\;\text{N·m}\) (clockwise). Hence the loop experiences a torque that tends to align its plane parallel to the field.

11. Summary Checklist (AO1)

  • Magnetic field lines: tangent = direction of B, density = magnitude, closed loops, no intersections.

  • Right‑hand grip rule → direction of B around a current.

  • Fleming’s left‑hand rule → direction of force on a current‑carrying conductor.

  • Magnetic flux \(\Phi = B A \cos\theta\) and Faraday’s law \(\varepsilon = -d\Phi/dt\) (Lenz’s law for direction).

  • Key formulas:

    \(B=\dfrac{\mu_{0}I}{2\pi r}\) (straight wire)

    \(B=\mu_{0} n I\) (ideal solenoid)

    \(F = q v B \sin\theta\) (moving charge)

    \(F = I L B \sin\theta\) (current element)

    \(\varepsilon = -\dfrac{d\Phi}{dt}\) (induction)

    \(\varepsilon = B\ell v\) (moving rod)

    \(v = \dfrac{E}{B}\) (velocity selector)

    \(\varepsilon_{\text{self}} = -L\,\dfrac{dI}{dt}\) (inductance)

  • Experimental visualisation: iron filings, compass grid, Hall probe, ferrofluid, search‑coil.

  • Extensions: quantum magnetic moments, MRI, transformer principle.

  • AO3 skills – set‑up, observe, record, evaluate magnetic‑field experiments.

12. Practice Questions (Mixed AO1–AO3)

  1. Diagramming (AO1): Sketch the magnetic field lines for a bar magnet. Label the north and south poles and indicate where the field is strongest.

  2. Calculation (AO2): A long straight wire carries 4 A. Find the magnetic field 5 cm from the wire and state its direction using the right‑hand grip rule.

  3. Conceptual (AO1): Explain why magnetic field lines never begin or end in empty space.

  4. Application (AO2): A particle of charge +2 µC moves at \(3\times10^{5}\,\text{m s}^{-1}\) perpendicular to a uniform magnetic field of 0.08 T. Calculate the magnitude of the magnetic force on the particle.

  5. Hall‑effect (AO3): In a Hall‑probe experiment a semiconductor plate of thickness 0.5 mm carries a current of 10 mA. The measured Hall voltage is 0.35 mV when a magnetic field of 0.20 T is applied. Determine the carrier concentration \(n\) (assume electron charge magnitude \(e = 1.6\times10^{-19}\,\text{C}\)).

  6. Velocity selector (AO2): An electric field of \(1.5\times10^{4}\,\text{V m}^{-1}\) is perpendicular to a magnetic field of 0.30 T. What speed of charged particles will pass through undeflected?

  7. Force on a current loop (AO2): A rectangular loop (0.12 m × 0.25 m) carries 2 A clockwise. It is placed in a uniform magnetic field of 0.40 T directed upward. Find the magnitude and direction of the net force on the loop and comment on any torque.

  8. Experimental design (AO3): Propose a method using iron filings and a sheet of paper to compare the field strengths of a solenoid and a bar magnet. Include how you would make the comparison quantitative.

  9. Extension – Toroid (AO1/AO2): Describe the magnetic field inside a toroid and explain why the external field is essentially zero. Calculate the field at a radius of 5 cm for a toroid with 500 turns carrying 3 A.

  10. Induction (AO2): A rectangular coil of 20 turns, area \(2.5\times10^{-3}\,\text{m}^2\), rotates at 60 rev s\(^{-1}\) in a uniform field of 0.30 T. Determine the peak induced e.m.f.