define magnetic flux density as the force acting per unit current per unit length on a wire placed at right- angles to the magnetic field

Force on a Current‑Carrying Conductor – Cambridge IGCSE/A‑Level (9702)

Learning Objective

Define magnetic flux density \(B\) as the force acting per unit current per unit length on a straight wire placed at right angles to a uniform magnetic field, and extend the definition to any angle between the wire and the field. Relate the concept to later syllabus topics (electromagnetic induction, AC machines) and to experimental determination of \(B\).

Key Concepts

• Magnetic field \(\mathbf{B}\) – a vector field that exerts a force on moving charges or on a current‑carrying conductor.
• Current \(I\) – the rate of flow of charge (conventional direction from positive to negative).
• Length of wire in the field \(L\) – the portion of the conductor that actually lies within the uniform magnetic region.
• Force \(\mathbf{F}\) – given by the Lorentz force law for a current element.

Vector Form of the Lorentz Force

For an infinitesimal element \(d\mathbf{l}\) of a current‑carrying wire the magnetic force is

\[

d\mathbf{F}=I\,d\mathbf{l}\times\mathbf{B}.

\]

For a straight segment of length \(L\) (with \(\mathbf{L}\) directed along the conventional current) this integrates to

\[

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

\]

The magnitude follows from the definition of the cross‑product:

\[

F = I\,L\,B\sin\theta,

\]

where \(\theta\) is the angle between \(\mathbf{L}\) and \(\mathbf{B}\). The scalar form is a direct consequence of the vector equation when \(\sin\theta = 1\) (wire ⟂ \(\mathbf{B}\)).

Definition of Magnetic Flux Density

For a wire perpendicular to the field (\(\theta = 90^{\circ}\)) the relationship simplifies to

\[

\boxed{B = \frac{F}{I\,L}}\qquad(\theta = 90^{\circ}).

\]

For any other orientation the general definition is

\[

\boxed{B = \frac{F}{I\,L\sin\theta}}.

\]

Units and Dimensions

Direction of the Force

The three vectors \(\mathbf{L}\), \(\mathbf{B}\) and \(\mathbf{F}\) are mutually perpendicular.

• Fleming’s left‑hand rule (conventional current):

  1. First finger → direction of \(\mathbf{B}\) (from north to south, into or out of the page).
  2. Second finger → direction of conventional current \(I\).
  3. Thumb → direction of the force \(\mathbf{F}\) on the conductor.

• Right‑hand rule for electron flow: because electrons move opposite to conventional current, point the thumb in the electron‑velocity direction, the fingers in \(\mathbf{B}\); the palm gives the direction of the force on a negative charge.

Link‑on to Other Syllabus Topics

The same cross‑product law governs the force on a single moving charge, \(\mathbf{F}=q\mathbf{v}\times\mathbf{B}\). This underpins:

• Electromagnetic induction – a changing magnetic flux (\(\Phi = B A\)) induces an emf (\(\mathcal{E}= -\dfrac{d\Phi}{dt}\)). The magnitude of \(B\) therefore directly influences the induced voltage in generators and transformers.
• AC machines (motors, generators) – the torque on a coil of \(N\) turns carrying current \(I\) in a field \(B\) is \(\tau = N I A B \sin\theta\); the basic force law is the building block for these devices.

Experimental Determination of \(B\)

A common laboratory method is the “current‑carrying wire in a known magnetic field” technique:

1. Place a straight segment of known length \(L\) in the uniform field of a calibrated magnet.
2. Pass a measured current \(I\) through the wire.
3. Measure the resulting force \(F\) with a force sensor or a torsion balance.
4. Calculate \(B\) using \(B = F/(I L)\) (or the \(\sin\theta\) form if the wire is not perpendicular).

Safety & Practical Considerations

Worked Example – Perpendicular Wire

Question: A straight wire 0.25 m long carries a current of 3.0 A. When the wire is placed perpendicular to a uniform magnetic field it experiences a force of 1.2 N. Find the magnetic flux density.

1. Use the definition for \(\theta = 90^{\circ}\):\(\displaystyle B = \frac{F}{I L}\).
2. Substitute the values:\(\displaystyle B = \frac{1.2\ \text{N}}{3.0\ \text{A}\times0.25\ \text{m}}.\)
3. Calculate:\(\displaystyle B = \frac{1.2}{0.75}=1.6\ \text{T}.\)

Answer: \(B = 1.6\ \text{tesla}\).

Worked Example – Wire at an Angle

Question: A 0.40 m long wire carries a current of 2.0 A. It is placed in a magnetic field of magnitude 0.80 T, making an angle of \(30^{\circ}\) with the field direction. Find the magnitude of the force on the wire.

1. Apply the general formula: \(\displaystyle F = I L B \sin\theta\).
2. Insert the data:\(\displaystyle F = 2.0\ \text{A}\times0.40\ \text{m}\times0.80\ \text{T}\times\sin30^{\circ}.\)
3. \(\sin30^{\circ}=0.5\), so \(\displaystyle F = 2.0\times0.40\times0.80\times0.5 = 0.32\ \text{N}.\)

Answer: The force is \(0.32\ \text{N}\) directed according to Fleming’s left‑hand rule.

Common Misconceptions & Clarifications

• “The magnetic field is a force.” – The field is a property of space; a force appears only when a charge or current interacts with the field.
• “\(F = I L B\) works for any orientation.” – This simple form is valid only for \(\theta = 90^{\circ}\). The correct general expression is \(F = I L B \sin\theta\).
• “Current direction equals electron flow.” – Conventional current is defined opposite to electron motion. Use Fleming’s left‑hand rule for conventional current and the right‑hand rule for electron flow.
• “The tesla is a unit of force.” – 1 T = 1 N·A⁻¹·m⁻¹; it measures magnetic field strength, not force.
• “The field is always uniform.” – In many practical situations the field varies with position; the formula above applies only to the portion of wire that experiences a uniform \(\mathbf{B}\). For non‑uniform fields the force must be integrated over the wire.

Summary

Magnetic flux density \(B\) quantifies the strength of a magnetic field at a point. It is defined by the relationship

\[

B = \frac{F}{I\,L}\qquad(\text{wire } \perp \mathbf{B}),

\]

or, more generally,

\[

B = \frac{F}{I\,L\sin\theta}.

\]

The SI unit is the tesla (T). The direction of the force follows Fleming’s left‑hand rule for conventional current, and the vectors \(\mathbf{L}\), \(\mathbf{B}\) and \(\mathbf{F}\) are mutually perpendicular. This principle underlies electromagnetic induction, the operation of AC machines, and the experimental determination of magnetic field strength.