Students will be able to:
The magnetic field \(\mathbf{B}\) at a point is defined as the force per unit positive charge moving with velocity \(\mathbf{v}\) at that point:
\[
\mathbf{B}= \frac{\mathbf{F}}{q\,v\sin\theta}\qquad\text{or}\qquad
\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}.
\]
For a straight conductor carrying current \(I\), point the thumb of the right hand in the direction of the conventional current; the curled fingers give the direction of the magnetic field lines encircling the wire.
Using Ampère’s law (or the Biot–Savart law) the magnitude of the field at a distance \(r\) from a long, straight wire is
\[
B=\frac{\mu_{0} I}{2\pi r},
\qquad\mu_{0}=4\pi\times10^{-7}\ \text{T·m·A}^{-1}.
\]
The Lorentz force on a charge \(q\) moving with velocity \(\mathbf{v}\) in a magnetic field \(\mathbf{B}\) is
\[
\mathbf{F}=q\,\mathbf{v}\times\mathbf{B},
\qquad
F = qvB\sin\theta .
\]
\[
I = n q A v_{d},
\]
where \(n\) = charge‑carrier number density (m\(^{-3}\)), \(q\) = charge of each carrier (C), \(A\) = cross‑sectional area (m\(^2\)), \(v_{d}\) = drift speed (m s\(^{-1}\)).
\[
N = n A L .
\]
\[
\mathbf{f}=q\,\mathbf{v}_{d}\times\mathbf{B}.
\]
\[
\mathbf{F}=N\mathbf{f}
=(nAL)q\,\mathbf{v}_{d}\times\mathbf{B}
=(nqAv_{d})\,L\,\hat{\mathbf{I}}\times\mathbf{B}
=I\,L\,\hat{\mathbf{I}}\times\mathbf{B}.
\]
\(\hat{\mathbf{I}}\) is a unit vector in the direction of the conventional current.
Magnitude:
\[
F = I L B \sin\theta .
\]
Special case required by the syllabus (wire ⟂ field): \(\theta = 90^{\circ}\) ⇒ \(\sin\theta = 1\), so
\[
F_{\perp}= I L B .
\]
When a current‑carrying conductor is placed in a magnetic field, the magnetic force on the charge carriers builds up a transverse electric field \(E_{H}\). At equilibrium
\[
qE{H}=qv{d}B \;\;\Longrightarrow\;\; E{H}=v{d}B .
\]
The measurable Hall voltage across a plate of width \(w\) is
\[
V{H}=E{H}w = v_{d} B w .
\]
Since \(I = n q A v_{d}\), the Hall voltage can also be written as
\[
V_{H}= \frac{I B w}{n q A}.
\]
Problem: A rectangular copper strip (width \(w = 2.0\ \text{mm}\), thickness \(t = 0.5\ \text{mm}\)) carries a current \(I = 3.0\ \text{A}\). It is placed in a uniform magnetic field \(B = 0.40\ \text{T}\) directed into the page. The free‑electron density in copper is \(n = 8.5\times10^{28}\ \text{m}^{-3}\). Find the Hall voltage across the width of the strip.
\[
V_{H}= \frac{(3.0)(0.40)(2.0\times10^{-3})}
{(8.5\times10^{28})(1.60\times10^{-19})(1.0\times10^{-6})}
\approx 1.8\times10^{-6}\ \text{V}=1.8\ \mu\text{V}.
\]
\[
B = \frac{\mu_{0} I}{2\pi r},
\qquad\text{direction given by the right‑hand rule (thumb = current).}
\]
For a tightly wound solenoid of \(n\) turns per unit length carrying current \(I\):
\[
B = \mu_{0} n I,
\]
field lines are parallel to the axis; the direction follows the right‑hand rule (curl fingers in the sense of the winding, thumb points along \(\mathbf{B}\)).
\[
B = \frac{\mu_{0} N I}{2\pi r},
\]
where \(N\) is the total number of turns and \(r\) the mean radius.
| Configuration | Formula for \(|\mathbf{B}|\) | Direction rule |
|---|---|---|
| Long straight wire | \(\displaystyle B=\frac{\mu_{0}I}{2\pi r}\) | Right‑hand thumb = current |
| Solenoid (length ≫ diameter) | \(\displaystyle B=\mu_{0} n I\) | Right‑hand curl = current loops |
| Toroid (circular coil) | \(\displaystyle B=\frac{\mu_{0} N I}{2\pi r}\) | Right‑hand thumb = axis of toroid |
Problem: A horizontal wire 0.30 m long carries a current of 5.0 A to the east. It is placed in a uniform magnetic field of magnitude 0.80 T directed vertically upward. Find the magnitude and direction of the force.
Problem: A rectangular coil of 4 turns has dimensions \(l = 0.10\ \text{m}\) and \(w = 0.05\ \text{m}\). It carries a current of 2.0 A and is placed in a uniform magnetic field of 0.60 T. The field makes an angle \(\phi = 30^{\circ}\) with the normal to the coil. Find the torque.
\[
\tau = (4)(2.0)(5.0\times10^{-3})(0.60)\sin30^{\circ}
= 1.2\times10^{-2}\ \text{N·m}.
\]
| Quantity | Symbol | Expression | Units |
|---|---|---|---|
| Current | I | I = n q A v_{d} | A (ampere) |
| Magnetic field (definition) | \(\mathbf{B}\) | \(\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}\) | T (tesla) |
| Field of a straight wire | \(B\) | \(B=\dfrac{\mu_{0} I}{2\pi r}\) | T |
| Field inside a solenoid | \(B\) | \(B=\mu_{0} n I\) | T |
| Force on a straight conductor | \(\mathbf{F}\) | \(\mathbf{F}= I L\,\hat{\mathbf{I}}\times\mathbf{B}\) | N |
| Magnitude of force | F | \(F = I L B \sin\theta\) | N |
| Torque on a coil (N turns) | \(\tau\) | \(\tau = N I A B \sin\phi\) | N·m |
| Hall voltage | V{H} | \(V{H}= \dfrac{I B w}{n q A}\) | V |
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