Force on a Current‑Carrying Conductor and the Use of Crossed Fields for Velocity Selection
1. Cambridge 9702 Syllabus – Topics Covered
- 18‑1/‑2 Electric fields and potential difference
- 19‑1 Capacitance and energy stored in a capacitor
- 20‑2 Force on a current‑carrying conductor
- 20‑3 Force on a moving charge (Lorentz force)
- 20‑4 Magnetic field of a current‑carrying wire (Biot‑Savart, Ampère’s law)
- 20‑5 Electromagnetic induction (Faraday’s & Lenz’s laws)
- 21‑1 Characteristics of AC (r.m.s. values, power)
- 21‑2 Rectification and smoothing
- 22‑1/‑2 Photo‑electric effect and quantum ideas (brief link)
- 23‑1/‑2 Nuclear physics (mass spectrometer principle)
- 24‑1/‑2 Medical physics (MRI, PET, radiation therapy)
2. Magnetic‑Field Refresher
- Magnetic flux density \( \mathbf{B} \) – measured in tesla (T). Direction is given by the direction a north‑pole needle points.
- Right‑hand rule (conventional current): thumb = direction of current (or velocity of a positive charge); fingers curl in the direction of the magnetic field produced.
- Uniform fields are produced by Helmholtz coils or the gap between the poles of an electromagnet.
- Field of a straight current‑carrying wire (Biot‑Savart)
\[
B = \frac{\mu_0 I}{2\pi r}
\]
where \(r\) is the radial distance from the wire and \(\mu_0 = 4\pi\times10^{-7}\,\text{T·m·A}^{-1}\).
- Solenoid (long coil) – inside the coil the field is approximately uniform:
\[
B = \mu_0 n I
\]
with \(n\) = number of turns per metre.
- Ampère’s law (integral form) – for any closed loop:
\[
\oint \mathbf{B}\cdot d\mathbf{l}= \mu_0 I_{\text{enc}}
\]
3. Electric‑Field Refresher
- Electric field \( \mathbf{E} \) – force per unit positive charge: \(\mathbf{E}= \mathbf{F}/q\) (units V m⁻¹).
- Uniform field between parallel plates of separation \(d\) and potential difference \(V\):
\[
E = \frac{V}{d}
\]
- Potential energy change for a charge \(q\) moving through the field: \( \Delta U = qV \).
4. Lorentz Force – General Form
The total electromagnetic force on a charge \(q\) moving with velocity \(\mathbf{v}\) in simultaneous electric and magnetic fields is
\[
\mathbf{F}=q\mathbf{E}+q\mathbf{v}\times\mathbf{B}
\]
- Electric part: \(\mathbf{F}_E = q\mathbf{E}\) (parallel to \(\mathbf{E}\)).
- Magnetic part: \(\mathbf{F}_B = qvB\sin\theta\) (perpendicular to both \(\mathbf{v}\) and \(\mathbf{B}\)).
5. Force on a Current‑Carrying Conductor
Derivation (from the Lorentz force on charge carriers)
- Number density of carriers: \(n\) (m⁻³); charge of each carrier: \(q\).
- Cross‑sectional area of the wire: \(A\); drift velocity of carriers: \(v_d\).
- Force on one carrier: \(\mathbf{f}=q\mathbf{v}_d\times\mathbf{B}\).
- Total carriers in a segment of length \(L\): \(N=nAL\).
- Total magnetic force: \(\mathbf{F}=N\mathbf{f}=nAL\,q\,\mathbf{v}_d\times\mathbf{B}\).
- Current definition: \(I=nqAv_d\) ⇒ \(nqAv_d = I\).
- Substituting gives the familiar vector result:
\[
\boxed{\mathbf{F}=I\mathbf{L}\times\mathbf{B}}
\]
Key Points
- Direction follows the right‑hand rule for \(\mathbf{I}\mathbf{L}\times\mathbf{B}\).
- Magnitude: \(F = I L B \sin\theta\) (θ = angle between wire direction and \(\mathbf{B}\)).
- For curved conductors, apply the formula to each infinitesimal element \(d\mathbf{L}\) and integrate.
6. Hall Effect – Demonstrating the Balance of Electric & Magnetic Forces
- When a current‑carrying slab is placed in a magnetic field, charge carriers are deflected sideways, creating a transverse electric field \(\mathbf{E}_H\) (the Hall field).
- Equilibrium condition:
\[
q\mathbf{v}_d\times\mathbf{B}+q\mathbf{E}_H = 0 \;\Longrightarrow\; E_H = v_d B
\]
- Hall voltage across the width \(w\) of the slab:
\[
V_H = E_H w = v_d B w = \frac{IB}{nq t}
\]
where \(t\) is the thickness of the conductor.
- Hall probes use this relationship to measure \(\mathbf{B}\); a typical Paper 3 practical asks you to calibrate a Hall probe with a known current.
7. Velocity Selector – Using Crossed Electric and Magnetic Fields
Principle of Operation
- Generate a uniform electric field \(\mathbf{E}\) (horizontal) and a uniform magnetic field \(\mathbf{B}\) (into the page) so that \(\mathbf{E}\perp\mathbf{B}\).
- Inject a beam of charged particles (charge \(q\)) with initial velocity \(\mathbf{v}\) perpendicular to both fields.
- For a particle moving to the right, the forces are:
- \(\mathbf{F}_E = q\mathbf{E}\) (upward for \(q>0\)).
- \(\mathbf{F}_B = q\mathbf{v}\times\mathbf{B}\) (downward for the chosen geometry).
- Zero net deflection requires \(\mathbf{F}_E + \mathbf{F}_B = 0\) ⇒
\[
qE = qvB \;\Longrightarrow\; \boxed{v = \frac{E}{B}}
\]
- Particles with \(v>E/B\) are deflected toward the magnetic side; those with \(v
Typical Laboratory Procedure (Paper 3 style)
- Set up parallel plates 5 mm apart; apply a variable voltage \(V\) to create \(E = V/d\).
- Place a pair of Helmholtz coils around the region; measure \(B\) with a calibrated Hall probe.
- Align an electron gun so the beam enters perpendicular to both fields.
- Adjust \(V\) (or the coil current) until the beam passes straight through a downstream slit or detector – this is the selected velocity.
- Record \(E\) and \(B\) and compute \(v = E/B\). Repeat for several \(B\) values to verify the linear relationship.
Example Calculation
Plates 5 mm apart, \(V = 500\;\text{V}\) → \(E = 1.0\times10^{5}\;\text{V m}^{-1}\).
Helmholtz coils give \(B = 0.20\;\text{T}\).
Selected speed:
\[
v = \frac{E}{B} = \frac{1.0\times10^{5}}{0.20}=5.0\times10^{5}\;\text{m s}^{-1}
\]
8. Electromagnetic Induction – Faraday’s & Lenz’s Laws
- Magnetic flux through a coil of \(N\) turns: \(\Phi = N\,\mathbf{B}\cdot\mathbf{A}\) (weber, Wb).
- Faraday’s law:
\[
\boxed{\mathcal{E}= -\frac{d\Phi}{dt}}
\]
The induced emf \(\mathcal{E}\) equals the negative rate of change of flux.
- Lenz’s law – the induced current creates a magnetic field that opposes the change in flux (the minus sign in Faraday’s law).
- Simple coil experiment (Paper 3): rotate a rectangular coil in a uniform magnetic field, measure the peak emf with a galvanometer, and verify \(\mathcal{E}=NBA\omega\sin\omega t\).
- Link to the velocity selector: after selection, ions enter a uniform magnetic field that forces them onto a circular path of radius
\[
r = \frac{mv}{qB}
\]
The changing magnetic flux through a surrounding detection coil induces an emf proportional to the ion’s speed – the principle of a mass spectrometer.
9. Alternating Current – RMS, Power, Rectification & Smoothing
- RMS current for a sinusoid: \(I_{\text{rms}} = I_{\text{peak}}/\sqrt{2}\). The same relation holds for voltage.
- Average power in a resistor:
\[
P_{\text{av}} = I_{\text{rms}}^{2}R = V_{\text{rms}}^{2}/R
\]
- Full‑wave rectifier (bridge of four diodes) converts the AC output of a velocity‑selector experiment into a unidirectional current that can be measured with a galvanometer.
- Smoothing (filter) capacitor: a capacitor \(C\) placed across the rectified output reduces the ripple voltage \(V_{\text{ripple}}\) according to
\[
V_{\text{ripple}} \approx \frac{I_{\text{load}}}{fC}
\]
where \(f\) is the ripple frequency (twice the line frequency for a full‑wave bridge).
- These concepts are frequently tested in Paper 3 practical questions involving AC circuits and signal conditioning.
10. Capacitors – Energy Stored in an Electric Field
- Capacitance \(C = Q/V\) (farads, F). For a parallel‑plate capacitor:
\[
C = \varepsilon_0\frac{A}{d}
\]
where \(A\) is plate area and \(d\) the separation.
- Energy stored:
\[
U = \frac{1}{2}CV^{2}= \frac{1}{2}QV = \frac{1}{2}\varepsilon_0 E^{2}A d
\]
showing the direct link between the electric field and stored energy.
- In the Thomson e/m experiment a capacitor is used to accelerate electrons; the kinetic energy gained is \(eV\), which is then related to the magnetic deflection.
11. Medical‑Physics Connections (Brief)
- MRI (Magnetic Resonance Imaging) – relies on strong, uniform static magnetic fields (\(B\approx1\!-\!3\;\text{T}\)) and radio‑frequency electric fields to manipulate nuclear spin.
- PET (Positron Emission Tomography) – detects \(\gamma\)-rays from annihilation events; magnetic fields are used in the detector electronics to guide charged particles.
- Radiation therapy – electron accelerators use crossed electric and magnetic fields to select electron energies before they strike the tumour.
12. Key Equations Summary
| Quantity |
Formula |
Units |
| Magnetic field of a long straight wire |
\(B = \dfrac{\mu_0 I}{2\pi r}\) |
T |
| Magnetic field inside a solenoid |
\(B = \mu_0 n I\) |
T |
| Force on a current‑carrying conductor |
\(\mathbf{F}= I\mathbf{L}\times\mathbf{B}\) |
N |
| Lorentz force (general) |
\(\mathbf{F}= q\mathbf{E}+ q\mathbf{v}\times\mathbf{B}\) |
N |
| Velocity selected by crossed fields |
\(v = \dfrac{E}{B}\) |
m s⁻¹ |
| Hall voltage |
\(V_H = \dfrac{IB}{nqt}\) |
V |
| Induced emf (Faraday) |
\(\mathcal{E}= -\dfrac{d\Phi}{dt}\) |
V |
| Radius of curvature in a magnetic field |
\(r = \dfrac{mv}{qB}\) |
m |
| RMS current (sinusoid) |
\(I_{\text{rms}} = I_{\text{peak}}/\sqrt{2}\) |
A |
| Capacitance of parallel plates |
\(C = \varepsilon_0 \dfrac{A}{d}\) |
F |
| Energy stored in a capacitor |
\(U = \dfrac{1}{2}CV^{2}\) |
J |
13. Quick Revision Checklist
- Define \(\mathbf{E}\) and \(\mathbf{B}\); state their units and how they are produced.
- Write the Lorentz force and identify the electric and magnetic components.
- Derive \( \mathbf{F}=I\mathbf{L}\times\mathbf{B} \) from charge‑carrier motion.
- Explain the Hall effect and give the expression for \(V_H\).
- State the condition for zero net force in a velocity selector and calculate \(v = E/B\).
- Recall Faraday’s law, Lenz’s law, and apply them to a rotating‑coil experiment.
- Convert peak AC values to rms; write the power formula for a resistor.
- Describe a full‑wave bridge rectifier and the role of a smoothing capacitor.
- Calculate the capacitance and stored energy of a parallel‑plate capacitor.
- Identify one medical‑physics application for each of the following: static magnetic field, crossed fields, and induced emf.