20 Magnetic Fields – Cambridge A‑Level Physics (9702)
Learning Objectives
- Define a magnetic field and explain how it is represented with field‑line diagrams.
- Calculate magnetic flux density B and magnetic flux Φ.
- State and use the Lorentz‑force law for a moving charge and the force law for a current‑carrying conductor (outcome 20.2).
- Apply the right‑hand rule correctly for both charge motion (positive charge) and conventional current (outcome 20.3).
- Describe the basic principles of electromagnetic induction (Faraday’s & Lenz’s laws) and relate them to common devices (outcome 20.5).
- Identify and correct common misconceptions.
20.1 Conceptual Background – What Is a Magnetic Field?
20.1.1 Definition of B
A magnetic field B is a region of space in which a moving electric charge experiences a magnetic force. B is a vector quantity called the magnetic flux density and is measured in tesla (T).
20.1.2 Field‑Line Diagrams
- Field lines emerge from the north pole of a magnet and enter the south pole.
- The density of lines (lines per unit area) indicates the magnitude of B.
- Field lines are continuous; they never start or stop in free space.
- The tangent to a field line at any point gives the direction of B at that point.
20.1.3 Magnetic Flux Density (B)
The direction of B is the direction a tiny north‑pole test magnet would point if placed at that location. Its magnitude is defined from the magnetic force on a charge:
\[
B = \frac{F}{qv\sin\theta}
\]
where F is the magnetic force on a charge q moving with speed v at an angle θ to the field.
20.1.4 Magnetic Flux (Φ)
\[
\Phi = \int \mathbf{B}\!\cdot\! d\mathbf{A}=BA\cos\theta
\]
- A = area of the surface.
- θ = angle between B and the outward normal to the surface.
- Unit: weber (Wb).
Link to later sections: Φ appears in Faraday’s law (Section 20.5) and in the torque expression for a current‑carrying coil (τ = NIAB sin θ).
20.2 Force on a Moving Charge – Lorentz Force
20.2.1 Lorentz‑Force Law
\[
\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}
\]
- Magnitude: \(F = qvB\sin\theta\).
- Direction: use the right‑hand rule for a positive charge (see 20.2.2). For a negative charge the force is opposite to the thumb.
20.2.2 Right‑Hand Rule for a Positive Charge
- Point your fingers in the direction of the velocity vector \(\mathbf{v}\).
- Sweep (or curl) your fingers toward the direction of the magnetic field \(\mathbf{B}\).
- Your thumb now points in the direction of the magnetic force \(\mathbf{F}\) on a positive charge.
For an electron (negative charge) simply reverse the thumb direction.
20.2.3 Worked Example – Proton in a Uniform Field
Problem: A proton (charge \(+e\)) moves at \(v=2.0\times10^{6}\ \text{m s}^{-1}\) perpendicular to a uniform magnetic field of magnitude \(B=0.50\ \text{T}\). Find the magnitude and direction of the force.
- Magnitude: \(F = qvB = (1.60\times10^{-19})(2.0\times10^{6})(0.50)=1.6\times10^{-13}\ \text{N}\).
- Direction: With \(\mathbf{v}\) to the right and \(\mathbf{B}\) into the page, the right‑hand rule gives \(\mathbf{F}\) upward.
20.3 Force on a Current‑Carrying Conductor
20.3.1 From Microscopic to Macroscopic – Derivation
- Force on a single charge carrier: \(\mathbf{F}q = q\,\mathbf{v}d\times\mathbf{B}\).
- Current definition: \(I = nqAvd\) (where n = carriers per unit volume, A = cross‑section, \(\mathbf{v}d\) = drift velocity).
- Number of carriers in a length \(L\): \(N = nAL\).
- Total force on all carriers:
\[
\mathbf{F}{\text{total}} = N q\,\mathbf{v}d\times\mathbf{B}
= (nAL)q\,\mathbf{v}_d\times\mathbf{B}
= I\,L\,\hat{\mathbf{I}}\times\mathbf{B}
= I\,\mathbf{L}\times\mathbf{B}
\]
This is the macroscopic force law used in the syllabus (outcome 20.2).
20.3.2 Vector Form
\[
\boxed{\mathbf{F}=I\,\mathbf{L}\times\mathbf{B}}
\]
- \(\mathbf{L}\) = vector of magnitude equal to the length of the wire, directed along the conventional current (from + to –).
- \(\mathbf{B}\) = magnetic flux density.
- \(\mathbf{F}\) = force on the wire (direction of the mechanical push on the conductor).
20.3.3 Right‑Hand Rule for \(\mathbf{L}\times\mathbf{B}\)
- Stretch the fingers of your right hand in the direction of the conventional current (\(\mathbf{L}\)).
- Rotate the hand so that you can sweep the fingers toward the direction of \(\mathbf{B}\).
- Your thumb points in the direction of the magnetic force \(\mathbf{F}\) on the wire.
- If the charge carriers are electrons, the force on the electrons is opposite to the thumb, but the wire itself moves in the thumb direction because the lattice is pulled opposite to the electron motion.
20.3.4 Direction Table (Common Orientations)
| Current \(\mathbf{L}\) (or \(\mathbf{I}\)) | Magnetic Field \(\mathbf{B}\) | Resulting Force \(\mathbf{F}\) |
|---|
| Into the page (×) | Right (+x) | Up (+y) |
| Right (+x) | Up (+y) | Into the page (×) |
| Up (+y) | Into the page (×) | Left (–x) |
20.3.5 Worked Example – Straight Wire in a Uniform Field
Problem: A straight wire of length \(L=0.30\ \text{m}\) carries a current \(I=5.0\ \text{A}\) to the north. It is placed in a uniform magnetic field of magnitude \(B=0.20\ \text{T}\) directed eastwards. Find the magnitude and direction of the magnetic force on the wire.
- Identify vectors:
- \(\mathbf{L}\) (or \(\mathbf{I}\)) → north (\(+\mathbf{j}\)).
- \(\mathbf{B}\) → east (\(+\mathbf{i}\)).
- Magnitude (since \(\theta=90^{\circ}\)):
\[
F = I L B = (5.0)(0.30)(0.20)=0.30\ \text{N}
\]
- Direction – right‑hand rule:
- Fingers → north (current).
- Rotate fingers toward east (field).
- Thumb points out of the page (toward the observer).
20.4 Torque on a Current‑Carrying Coil (Magnetic Dipole)
For a rectangular coil of \(N\) turns, each of area \(A\), carrying current \(I\) in a uniform field \(\mathbf{B}\):
\[
\boxed{\tau = NIAB\sin\theta}
\]
- \(\theta\) = angle between the normal to the coil and \(\mathbf{B}\).
- The torque tends to align the coil’s magnetic moment \(\boldsymbol{\mu}=NI\mathbf{A}\) with the field.
This expression underpins the operation of electric motors (outcome 20.4) and provides a bridge to electromagnetic induction (Section 20.5).
20.5 Electromagnetic Induction
20.5.1 Faraday’s Law
\[
\mathcal{E} = -\frac{d\Phi}{dt}
\]
- \(\mathcal{E}\) = induced emf (V).
- \(\Phi\) = magnetic flux through the loop.
- The minus sign embodies Lenz’s law.
20.5.2 Lenz’s Law
The induced current flows in a direction that creates a magnetic field opposing the change in flux that produced it. This guarantees energy conservation.
20.5.3 Example – Rotating Coil (Paper 5 Skill)
- Coil area \(A\) rotates in a uniform field \(B\) with angular speed \(\omega\).
- Flux varies as \(\Phi(t)=BA\cos\omega t\).
- Induced emf:
\[
\mathcal{E}(t)= -\frac{d\Phi}{dt}=BA\omega\sin\omega t
\]
The peak emf is \(\mathcal{E}_{\text{max}} = BA\omega\).
20.5.4 Practical Devices
- Electric motor: Torque \(\tau = NIAB\sin\theta\) produces rotation.
- Generator: Mechanical rotation of a coil changes \(\Phi\), inducing an emf.
- Hall probe: A current‑carrying semiconductor develops a transverse Hall voltage \(V_H = \dfrac{IB}{nqt}\) that measures B.
- Magnetic levitation, loudspeakers, and cathode‑ray tubes also rely on \(\mathbf{F}=q\mathbf{v}\times\mathbf{B}\) or \(\mathbf{F}=I\mathbf{L}\times\mathbf{B}\).
20.6 Common Misconceptions & How to Avoid Them
| Misconception | Correct Understanding |
|---|
| Electron vs. conventional current | Right‑hand rule always uses the direction of conventional current. For electrons, reverse the resulting force direction. |
| Zero force condition | When \(\mathbf{v}\) (or \(\mathbf{I}\)) is parallel or antiparallel to \(\mathbf{B}\), \(\sin\theta=0\) → no magnetic force. |
| Force on the wire vs. on individual charges | \(\mathbf{F}=I\mathbf{L}\times\mathbf{B}\) is the vector sum of the microscopic forces on all charge carriers; the lattice feels the opposite reaction. |
| Direction of induced emf | Remember the negative sign in Faraday’s law; Lenz’s law tells you the sense of the induced current. |
| “Magnetic field lines” are physical objects | They are a visual aid; the real field is described by the vector \(\mathbf{B}\) at every point. |
20.7 Practice Questions
A rectangular loop carries a current \(I\) clockwise when viewed from above. The loop lies in a uniform magnetic field directed into the page.
- a) State the direction of the magnetic force on each side of the loop.
- b) Explain why the net force on the whole loop is zero, yet a torque is produced.
A proton moves with speed \(2.0\times10^{6}\ \text{m s}^{-1}\) perpendicular to a magnetic field of \(0.5\ \text{T}\). Calculate the magnitude of the magnetic force. (Charge of a proton \(=+e\)).
Explain why a current‑carrying wire placed parallel to a magnetic field experiences no magnetic force.
A coil of 200 turns, each of area \(5.0\times10^{-3}\ \text{m}^{2}\), is rotated at \(60\ \text{rev s}^{-1}\) in a magnetic field of \(0.30\ \text{T}\). Determine the peak induced emf.
Using the right‑hand rule, determine the direction of the force on an electron moving northward through a magnetic field that points upward.
20.8 Key Take‑aways
- The magnetic force on a moving charge is \(\mathbf{F}=q\mathbf{v}\times\mathbf{B}\); on a conductor it is \(\mathbf{F}=I\mathbf{L}\times\mathbf{B}\).
- Direction is obtained with the right‑hand rule using conventional current (or positive charge velocity).
- The force is maximal when \(\mathbf{v}\) (or \(\mathbf{I}\)) is perpendicular to \(\mathbf{B}\) and vanishes when they are parallel.
- Magnetic flux \(\Phi = BA\cos\theta\) links the field to an area; a changing flux induces an emf according to Faraday’s law.
- Lenz’s law gives the sense of the induced current and guarantees energy conservation.
- These principles underpin everyday technology: electric motors, generators, Hall‑effect sensors, magnetic levitation, and particle‑deflection devices.