describe the motion of a charged particle moving in a uniform magnetic field perpendicular to the direction of motion of the particle
Learning Objective
Describe the motion of a charged particle moving in a uniform magnetic field that is perpendicular to its velocity, and use this description to derive and apply the magnetic‑force formula for a straight current‑carrying conductor.
1. Magnetic Field – Concept and Direction
- Definition: A magnetic field is a region of space in which a moving charge (or a current‑carrying conductor) experiences a magnetic force.
- Magnetic field lines: Imaginary lines drawn such that the tangent to a line at any point gives the direction of the magnetic field vector \(\mathbf B\) at that point. The density of lines indicates the magnitude of \(\mathbf B\).
- Right‑hand grip rule (field of a straight current): Point the thumb of the right hand in the direction of conventional current \(I\); the curled fingers give the direction of the magnetic field lines that circle the wire.
2. Magnetic Flux Density (\( \mathbf B \))
- Definition (force per unit current per unit length):
\[
B=\frac{F}{IL\sin\theta}\qquad\left[\text{unit: tesla (T)}\right]
\]
where \(F\) is the magnetic force on a straight wire of length \(L\) carrying current \(I\), and \(\theta\) is the angle between the wire (current direction) and the field.
- \(\mathbf B\) is a vector quantity; its direction is given by the field‑line convention described above.
3. Lorentz Force on a Single Charge
The magnetic force on a charge \(q\) moving with velocity \(\mathbf v\) in a magnetic field \(\mathbf B\) is
\[
\mathbf F = q\,\mathbf v \times \mathbf B
\]
- Magnitude: \(F = qvB\sin\phi\), where \(\phi\) is the angle between \(\mathbf v\) and \(\mathbf B\).
- Direction: given by the right‑hand rule for a positive charge (thumb = \(\mathbf v\), fingers = \(\mathbf B\), palm pushes in the direction of \(\mathbf F\)). For a negative charge the force is opposite to the palm direction.
4. Uniform Circular Motion of a Charged Particle (\(\mathbf v\perp\mathbf B\))
When the velocity is perpendicular to the magnetic field (\(\phi = 90^{\circ}\)) the magnetic force is always at right angles to the motion and therefore acts as a centripetal force.
\[
qvB = \frac{mv^{2}}{r}\qquad\Longrightarrow\qquad
r = \frac{mv}{qB}
\]
- Radius of the path: proportional to the particle’s momentum \(mv\) and inversely proportional to the product \(qB\).
- Cyclotron (angular) frequency:
\[
\omega = \frac{v}{r}= \frac{qB}{m}
\]
Note that \(\omega\) (and the linear frequency \(f=\omega/2\pi\)) depends only on the charge‑to‑mass ratio and the magnetic field – it is independent of the speed.
- Direction of motion: Use the right‑hand rule for the force; the particle continuously turns, tracing a circle in a plane perpendicular to \(\mathbf B\).
5. From a Single Charge to a Current‑Carrying Conductor
In a conductor the charge carriers (usually electrons) drift with speed \(v_{d}\). For a wire of length \(L\), cross‑sectional area \(A\) and carrier number density \(n\):
\[
I = nqAv_{d}\qquad\text{(definition of current in terms of drift speed)}
\]
The total magnetic force on all carriers in the segment is
\[
F{\text{total}} = (nqAv{d})\,L\,B\sin\theta = I L B\sin\theta
\]
This leads directly to the general magnetic‑force formula for a straight conductor.
6. Magnetic Force on a Straight Conductor
\[
\boxed{F = B I L \sin\theta}
\]
- \(\theta\) = angle between the direction of conventional current and the magnetic field.
- Maximum force when \(\theta = 90^{\circ}\) ( \(F = BIL\) ).
- Zero force when the wire is parallel to the field (\(\theta = 0^{\circ}\)).
7. Fleming’s Left‑Hand Rule
Place the left hand so that the first finger points along \(\mathbf B\) and the second finger along the direction of current \(I\); the thumb then points in the direction of the force \(\mathbf F\) on the wire.
8. Magnetic Field Produced by a Straight Conductor (Biot–Savart Law)
A current‑carrying straight wire generates a circular magnetic field around it. The magnitude at a distance \(r\) from the wire is
\[
B = \frac{\mu{0} I}{2\pi r}\qquad\left(\mu{0}=4\pi\times10^{-7}\,\text{T·m·A}^{-1}\right)
\]
- Direction of the field is given by the right‑hand grip rule (thumb = current, fingers = field lines).
- This completes the two‑way link required by the syllabus: a magnetic field exerts a force on a current, and a current creates a magnetic field.
9. Applications and Practical Skills
- Hall effect: When a current‑carrying conductor is placed in a magnetic field, the magnetic force separates charge carriers, producing a transverse Hall voltage. Hall probes exploit this to measure both magnitude and direction of \(\mathbf B\).
- Measuring magnetic force (lab skill): A known current and length of wire are placed between the poles of a calibrated electromagnet. The resulting force is measured with a spring balance, allowing verification of \(F = BIL\).
10. Summary Table
| Quantity | Symbol | Expression | Notes |
|---|---|---|---|
| Magnetic flux density | \(B\) | \(B=\dfrac{F}{IL\sin\theta}\) | unit = tesla (T) |
| Magnetic force on a charge | \(\mathbf F\) | \(q\,\mathbf v\times\mathbf B\) | right‑hand rule for \(q>0\) |
| Force magnitude ( \(v\perp B\) ) | \(F\) | \(qvB\) | |
| Radius of circular path | \(r\) | \(\displaystyle\frac{mv}{qB}\) | proportional to momentum |
| Cyclotron (angular) frequency | \(\omega\) | \(\displaystyle\frac{qB}{m}\) | independent of speed |
| Current (drift speed) | \(I\) | \(nqAv_{d}\) | \(n\)= carrier density |
| Force on a straight conductor | \(F\) | \(BIL\sin\theta\) | maximal at \(\theta=90^{\circ}\) |
| Magnetic field around a straight wire | \(B\) | \(\displaystyle\frac{\mu_{0}I}{2\pi r}\) | circular field lines |
11. Worked Examples
Radius of a proton’s path
\(m = 1.67\times10^{-27}\,\text{kg}\), \(q = +e = 1.60\times10^{-19}\,\text{C}\), \(v = 2.0\times10^{6}\,\text{m s}^{-1}\), \(B = 0.5\,\text{T}\) (perpendicular).
\[
r = \frac{mv}{qB}
= \frac{(1.67\times10^{-27})(2.0\times10^{6})}{(1.60\times10^{-19})(0.5)}
\approx 4.2\times10^{-2}\,\text{m}
\]
Force on a wire at an angle
\(L = 0.30\;\text{m}\), \(I = 5.0\;\text{A}\), \(B = 0.5\;\text{T}\), \(\theta = 30^{\circ}\).
\[
F = B I L \sin\theta
= (0.5)(5.0)(0.30)\sin30^{\circ}
= 0.375\;\text{N}
\]
Measuring \(B\) with a known force
\(L = 0.40\;\text{m}\), \(I = 2.0\;\text{A}\), measured force \(F = 0.80\;\text{N}\) (wire ⟂ \(\mathbf B\)).
\[
B = \frac{F}{IL} = \frac{0.80}{(2.0)(0.40)} = 1.0\;\text{T}
\]
12. Key Points to Remember
- The magnetic force on a moving charge is always perpendicular to both the velocity and the magnetic field.
- For \(\mathbf v\perp\mathbf B\) the charge follows a uniform circular path with radius \(r = mv/(qB)\) and angular frequency \(\omega = qB/m\).
- Magnetic flux density \(B\) is defined as the force per unit current per unit length: \(B = F/(IL\sin\theta)\).
- The force on a straight conductor is \(F = B I L \sin\theta\); direction given by Fleming’s left‑hand rule.
- A current‑carrying straight wire produces a circular magnetic field given by \(B = \mu_{0}I/(2\pi r)\); direction given by the right‑hand grip rule.
- The Hall effect provides a practical method of measuring magnetic fields by detecting the transverse voltage caused by the magnetic force on charge carriers.
