12.4 Force on a Current-Carrying Conductor

When a wire carrying an electric current is placed in an external magnetic field, it experiences a force. This phenomenon is a fundamental principle of electromagnetism and is the basis for the operation of electric motors, speakers, and many other devices.

1. The Origin of the Force

The force on the wire arises from the interaction between the external magnetic field and the magnetic field created by the moving charges (the current) within the wire.

• On one side of the wire, the two magnetic fields are in the same direction, and they add together to create a stronger field.
• On the other side of the wire, the fields are in opposite directions, and they partially cancel, creating a weaker field.
• This difference in magnetic field strength on either side of the wire results in a net force pushing the wire from the region of the stronger field to the region of the weaker field.

2. Formula for the Magnetic Force

The magnitude of the magnetic force ($F$) on a straight conductor of length $L$, carrying a current $I$, and placed in a uniform magnetic field of strength $B$, is given by:

$F=BIL\sin \theta$

Where:

• $F$ is the magnetic force (in Newtons, $\text{N}$).
• $B$ is the magnetic field strength (in Teslas, $\text{T}$).
• $I$ is the current in the wire (in Amperes, $\text{A}$).
• $L$ is the length of the wire that is inside the magnetic field (in meters, $\text{m}$).
• $\theta$ is the angle between the direction of the current and the direction of the magnetic field.

In vector form, the force is given by the cross product:

$\vec{F}=I(\vec{L}\times \vec{B})$

The direction of the vector $\vec{L}$ is the direction of the current flow. This relates to the behavior of Scalar and Vector Quantities→.

3. Maximum and Minimum Force

The magnitude of the force depends critically on the orientation of the wire with respect to the magnetic field.

• Maximum Force ($F_{\text{max}}$): The force is at its maximum when the wire is perpendicular to the magnetic field ($\theta =90^{\circ }$, so $\sin (90^{\circ })=1$).

$F_{\text{max}}=BIL$

• Minimum Force ($F_{\text{min}}$): The force is zero when the wire is parallel to the magnetic field ($\theta =0^{\circ }$ or $\theta =180^{\circ }$, so $\sin (0^{\circ })=0$).

$F_{\text{min}}=0$

4. Fleming's Left-Hand Rule

Fleming's Left-Hand Rule is a simple mnemonic used to determine the direction of the force on a current-carrying conductor in a magnetic field.

To use the rule, hold your left hand with the thumb, forefinger, and middle finger mutually perpendicular to each other:

• Forefinger: Points in the direction of the magnetic Field ($\vec{B}$).
• Middle Finger: Points in the direction of the Current ($\vec{I}$).
• Thumb: Points in the direction of the Force or Thrust ($\vec{F}$).

Possible Questions and Answers

Q: What is the underlying reason for the force on the wire?

A: The force on the wire is the macroscopic result of the sum of the individual magnetic forces (Lorentz forces) acting on each of the moving charge carriers (electrons) that make up the current within the wire.

Q: How does an electric motor work?

A: An electric motor uses this principle. It consists of a coil of wire placed in a magnetic field. When current flows through the coil, the magnetic force on the sides of the coil creates a turning effect (a torque). This torque causes the coil to rotate, converting electrical energy into rotational mechanical energy.