PhysicsLAB: Torque on a Current-Carrying Loop

Remember that when a current-carrying wire is placed in an external magnetic field then it will experience a magnetic force that can be calculated with the equation

and obeys the right hand rule.

thumb points in the direction of the current, I
fingers point in the direction of the external magnetic field, B
palm faces the direction of the force, F

This physlet by Walter Fendt illustrates this Lorentz force.

Example #1: Now let's place a freely-pivoting loop carrying a clockwise (red arrow) current in an external (+x) magnetic field.

as the current flows up the left side, it will experience a force in the -z direction.
as the current flows across the top of the loop, no force is exerted since the current and the magnetic field are parallel.
as the current flows down the right side, it will experience a force in the +z direction.

These forces will result in the right side of the loop rotating towards the reader.

Example #2: Now let's place the same freely-pivoting loop carrying a clockwise (red arrow) current in an external (+z) magnetic field.

as the current flows up the left side, it will experience a force in the +x direction.
as the current flows across the top of the loop, it will experience a force in the -y direction
as the current flows down the right side, it will experience a force in the -x direction.

Since the lines of action of both forces along the vertical sides pass through the axis of rotation they will not produce a torque. Note that the line of action of the force along the top section of the loop runs parallel to the axis and consequently can also not produce a torque. In this orientation, the coil will not rotate about the specified axis.

Every current-carrying coil has an area vector, A, that is oriented perpendicular to is cross-sectional area and points in the direction dictated by the right hand curl rule:


I circulates clockwise A points in the -z direction B points in the +x direction the right edge of the coil would rotate towards the reader	I circulates counter-clockwise A points in the +z direction B points in the +x direction the right edge of the coil would rotate away from the reader
aerial view	aerial view

Take a moment and investigate the following physlet modeling the rotation of the current-carrying loop in a magnetic field by Dr. Scott at Lawrence Technological University in Southfield, Michigan. Notice how the direction/magnitude of the current, direction/magnitude of the magnetic field and the size of the angle between the magnetic moment (area vector) affect the loop's rotation.

When the area vector is at right angles to the magnetic field the torque is maximized. Conversely, when the area vector is parallel to the magnetic field no torque is produced as evidenced in our second introductory example.

So how do we calculate the magnitude of the torque on a current-carrying coil?

Returning to our initial example,

We see that the torque

can be calculated using the appropriate values for r and F as

If there was more than one loop, the expression would be multiplied by the number of loops, N.

The expression NIA is called the magnetic moment of the loop and it measured in Am². Although we have derived this equation for a rectangular loop, it can be used with any planar loop of any geometry - in particular, circular loops whose areas are

Refer to the following information for the next three questions.

Suppose you have a circular loop of radius 0.25 meters that has 100 turns of wire. The coil carries 2 amps of current while in a magnetic field having a magnitude of 10 T.

	What is the coil's magnetic moment?

	What is the maximum torque the coil experiences?

	Which would have a greater effect: reducing the radius by a factor of 2 or reducing the number of loops by a factor of 2?

To see an application of the torque on a current-carrying loop, investigate this physlet by Walter Fendt of an electric DC motor.