20.2 Force on a Current-Carrying Conductor

2026 Syllabus Objectives

By the end of these notes, you should be able to:

  1. Understand that a force might act on a current-carrying conductor placed in a magnetic field.
  2. Recall and use the equation F = BIL sin θ, and use Fleming's left-hand rule to find the direction of the force.
  3. 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.

1. Why Does a Current-Carrying Conductor Experience a Force?

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A conductor (such as a copper wire) carrying an electric current produces its own magnetic field around it. When you place this conductor inside an external magnetic field (for example, between the poles of a magnet), the two magnetic fields interact with each other. The result of this interaction is a force acting on the conductor.

Key point: The force only arises because of the interaction between the conductor's own magnetic field and the external magnetic field. A wire with no current flowing through it will feel no magnetic force, even if it is sitting inside a magnetic field.

Demonstrating the Force

You can observe this force with a simple experiment:

  • Place a copper rod horizontally between the poles of a strong magnet so that the rod sits inside the magnetic field.
  • Connect the rod to a circuit so that current can flow through it.
  • When the current is switched on, the rod jumps or moves — it accelerates in the direction of the force.
  • If you reverse the direction of the current, the rod moves in the opposite direction.
  • If you flip the magnet so the field direction reverses, the rod again moves in the opposite direction.

This shows that the direction of the force depends on both the direction of the current and the direction of the magnetic field.

2. The Equation F = BIL sin θ

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The size (magnitude) of the force on a current-carrying conductor in a magnetic field is given by:

F=BILsin⁡θ\boxed{F = BIL \sin\theta}

What Each Letter Means

SymbolQuantityUnit
FForce on the conductorNewtons (N)
BMagnetic flux density (strength of the field)Tesla (T)
ICurrent through the conductorAmperes (A)
LLength of the conductor that is inside the fieldMetres (m)
θAngle between the conductor and the magnetic field linesDegrees (°)

Important: L is not the total length of the wire — it is only the length of wire that actually sits within the magnetic field.

How the Angle θ Affects the Force

The sin θ part of the equation tells us that the angle between the wire and the field lines matters a great deal.

Three important cases:

Case 1 — Wire perpendicular to the field (θ = 90°)

  • sin 90° = 1
  • Force is at its maximum: F = BIL
  • This gives the strongest push on the wire.

Case 2 — Wire at an angle θ to the field

  • Force is F = BIL sin θ
  • The force is somewhere between zero and the maximum.

Case 3 — Wire parallel to the field (θ = 0°)

  • sin 0° = 0
  • Force = zero
  • No force acts on the wire at all when it runs along the same direction as the field lines.

A simple summary table:

Angle θsin θForce F
90°1F = BIL (maximum)
30°0.5F = 0.5 BIL
0°0F = 0 (no force)

What Increases the Force?

From the equation F = BIL sin θ, you can see that the force on the conductor gets bigger if you:

  • Increase the magnetic flux density B (use a stronger magnet).
  • Increase the current I (push more current through the wire).
  • Increase the length L of wire inside the field.
  • Increase the angle θ towards 90° (make the wire more perpendicular to the field).

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