Determine the direction of the force on beams of charged particles in a magnetic field

4.5.4 Force on a Current‑Carrying Conductor

Learning Objective

Determine the direction of the magnetic force on a beam of charged particles or on a current‑carrying conductor when it moves through a magnetic field, and describe the classic experiment that demonstrates this force (including the effect of reversing the current and the magnetic field).

1. Theory – Magnetic Force

  • Single moving charge

    \[

    \mathbf{F}=q\,\mathbf{v}\times\mathbf{B}

    \]

    • \(q\) – charge (C)

    • \(\mathbf{v}\) – velocity vector (m s\(^{-1}\))

    • \(\mathbf{B}\) – magnetic‑field vector (T)

  • Straight conductor of length \(L\) carrying a current \(I\)

    \[

    \mathbf{F}=I\,\mathbf{L}\times\mathbf{B}

    \]

    \(\mathbf{L}\) points in the direction of conventional current (positive to negative).

  • Magnitude

    \[

    F = I L B \sin\theta

    \]

    \(\theta\) = angle between \(\mathbf{L}\) (or \(\mathbf{v}\)) and \(\mathbf{B}\).

2. Right‑Hand Rule (for a positive charge or conventional current)

  1. Stretch the fingers of your right hand in the direction of the velocity \(\mathbf{v}\) (or current \(\mathbf{I}\)).

  2. Rotate the hand so that, when you bend the fingers, they point in the direction of the magnetic field \(\mathbf{B}\).

  3. Extend the thumb – it points in the direction of the force \(\mathbf{F}\) on a positive charge.

  4. If the moving charge is negative, the force is opposite to the thumb direction.

Right‑hand rule showing v, B and F

Right‑hand rule: thumb = \(\mathbf{F}\), fingers = \(\mathbf{v}\) (or \(\mathbf{I}\)), palm‑facing = \(\mathbf{B}\).

3. Classic Experiment – Horseshoe‑Magnet & Current‑Carrying Wire

Apparatus

  • Horseshoe magnet (field \(\mathbf{B}\) directed into the page between the poles).

  • Straight insulated copper wire, length ≈ 10 cm, mounted on a low‑friction pivot so it can swing freely.

  • Battery (≈ 1.5 V) and a switch to control the current direction.

  • Optional: a small mirror attached to the wire and a laser pointer to measure the deflection angle.

Procedure

  1. Connect the wire to the battery so that a current flows from left to right (as drawn on the diagram).

  2. Observe the sideways deflection of the wire.

  3. Reverse the current (swap the battery leads) and note the opposite deflection.

  4. Flip the magnet (reverse \(\mathbf{B}\)) and observe that the deflection also reverses.

  5. Reverse both the current and the magnetic field; the wire returns to the original deflection direction.

Observations

  • The wire moves perpendicular to both \(\mathbf{I}\) and \(\mathbf{B}\) – exactly as predicted by the right‑hand rule.

  • Two reversals (current + field) give the same force direction as the original set‑up.

Sketch of the horseshoe‑magnet experiment

Horseshoe‑magnet set‑up: wire (blue), pivot (black dot), magnet poles (N, S), battery (red), and direction of \(\mathbf{I}\) and \(\mathbf{B}\).

Practical tip – measuring the force

  • Attach a lightweight mirror to the wire and shine a laser onto a distant screen; the spot displacement gives the deflection angle \(\alpha\).

  • Using the torque balance \(\tau = F d\) (where \(d\) is the distance from the pivot to the wire’s centre) you can calculate the force:

    \[

    F = \frac{m g d_{\text{cm}} \sin\alpha}{d}

    \]

    (with \(m\) the mass of the wire segment and \(g = 9.8\ \text{m s}^{-2}\)).

  • Safety: secure the wire to the pivot, avoid loose connections, and never touch the magnet or live wires with wet hands.

4. Relative Directions of \(\mathbf{I}\), \(\mathbf{B}\) and \(\mathbf{F}\)

CaseCurrent \(\mathbf{I}\)Magnetic Field \(\mathbf{B}\)Resulting Force \(\mathbf{F}\) (thumb direction)
(i)Original directionOriginal directionGiven by the right‑hand rule
(ii)Reversed (\(\mathbf{I}\rightarrow -\mathbf{I}\))OriginalForce reverses (opposite to case i)
(iii)OriginalReversed (\(\mathbf{B}\rightarrow -\mathbf{B}\))Force reverses (opposite to case i)
(iv)ReversedReversedForce returns to the direction of case i (two reversals cancel)

5. Determining the Force Direction – Worked Examples

Example 1: Beam of Protons

Problem: A beam of protons (\(q=+e\)) travels eastward with speed \(v\) and enters a uniform magnetic field that points vertically upward. What is the direction of the magnetic force on the protons?

Solution

  1. Vectors: \(\mathbf{v}\) – east; \(\mathbf{B}\) – upward.

  2. Right‑hand rule: fingers point east, bend upward, thumb points south.

  3. Protons are positively charged, so \(\mathbf{F}\) is southward.

Electron‑beam contrast

If the same beam were made of electrons (\(q=-e\)), the force would be opposite to the thumb direction. Using the right‑hand rule as above gives a thumb pointing south; the actual force on the electrons would therefore be northward.

Example 2: Straight Current‑Carrying Wire

Problem: A horizontal wire carries a current of 5 A from left to right. It is placed in a uniform magnetic field of 0.2 T that points into the page. Determine the direction of the magnetic force on the wire and calculate its magnitude if the wire segment inside the field is 0.10 m long.

Solution

  1. Vectors: \(\mathbf{I}\) – left → right; \(\mathbf{B}\) – into the page (× symbols).

  2. Right‑hand rule: fingers point right, bend into the page, thumb points upward.

  3. Force direction: upward (perpendicular to both \(\mathbf{I}\) and \(\mathbf{B}\)).

  4. Magnitude:

    \[

    F = I L B \sin 90^{\circ}= (5\ \text{A})(0.10\ \text{m})(0.2\ \text{T}) = 0.10\ \text{N}.

    \]

6. When the Magnetic Force Is Zero

If the velocity (or current) is parallel or antiparallel to the magnetic field (\(\mathbf{v}\parallel\mathbf{B}\) or \(\mathbf{I}\parallel\mathbf{B}\)), then \(\theta = 0^{\circ}\) or \(180^{\circ}\) and

\[

\sin\theta = 0\;\;\Longrightarrow\;\;\mathbf{F}=0.

\]

The charge (or conductor) moves straight through the field without any deflection.

7. Common Misconceptions

  • Current vs. electron flow: The right‑hand rule uses conventional current (direction of positive charge flow). For electron flow, reverse the current direction before applying the rule.

  • Sign of the charge: Negative charges experience a force opposite to the thumb direction.

  • Force always perpendicular? Only when \(\theta = 90^{\circ}\). If \(\theta = 0^{\circ}\) or \(180^{\circ}\) the force vanishes.

8. Quick Checklist for Determining the Force Direction

  1. Identify the sign of the moving charge (or adopt conventional current).

  2. Draw the velocity \(\mathbf{v}\) (or current \(\mathbf{I}\)) vector.

  3. Draw the magnetic‑field vector \(\mathbf{B}\).

  4. Apply the right‑hand rule – note the thumb direction.

  5. If the charge is negative, reverse the thumb direction.

  6. State the final direction of \(\mathbf{F}\) relative to your diagram.

9. Summary

The magnetic force on a moving charge or on a current‑carrying conductor is given by the cross‑product \(\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}\) (or \(\mathbf{F}=I\,\mathbf{L}\times\mathbf{B}\)). Its magnitude is \(F = I L B \sin\theta\). The direction is found with the right‑hand rule for positive charges (or conventional current) and reversed for negative charges. The horseshoe‑magnet experiment provides a clear, observable demonstration of the force and shows how reversing the current or the magnetic field reverses the force direction. Mastery of these concepts enables students to predict the behaviour of particle beams and conductors in magnetic fields – a core requirement of the Cambridge IGCSE Physics (0625) syllabus.