1. Magnetic Forces and Torques

2. Review: Charged Particle in an external field:
Straight wire of length L with current I in a uniform external field B:
Note: L points in direction of positive current flow

3. Force on a current-carrying wire (general case)
If B not uniform, and/or wire not straight: the force dF on a short segment of vector length dL is
dF = IdL x B
The total force on the wire is: dF B I
Segment of length dL dL I

4. Example 1 Find the force on: The straight wire The semicircular wire
The whole circuit y
x x x I R x B
(uniform)
For (b): start with force dF due to an infinitesimal piece, and do the integral.

5. Solution

6. Theorem: For any closed current loop in a uniform magnetic field,
Total magnetic force on the loop = 0
Proof:
(if B is a constant vector)
But: for a closed loop, so F = 0

7. Torque: Although there is no net force on a circuit in a uniform field, there may be a net torque. The torques due to equal and opposite forces applied at different locations do not necessarily cancel.
We can calculate the torque directly for a rectangular loop. There is also a simple rule, which actually applies to a plane loop of any shape. First we need to define the “magnetic dipole moment” (a vector) for a current loop.

8. Magnetic Dipole Moment
(or “magnetic moment”)
Define the magnetic moment of a small current loop by
current
Area of loop
Note the right-hand rule! – fingers follow I, thumb points in μ m

9. Torque on a Current Loop (Uniform B)
Example: a rectangular loop B A h C D w
Forces:
FAB= I w B sinθ
FAD= I h B
FBC= I h B x
FCD= I w B sinθ

10. Top view: FBC= I h B x I into the page I w w•sinθ FAD = I h B
Torque (about any pivot; e.g., at A)

11. The torque on a current loop with magnetic dipole moment  in a uniform external field is:
Where μ = IA

12. Example 2
Find the torque on a flat, horizontal, circular coil due to the magnetic field of the Earth.
50o
north
0.5 x 10-4 T
Circular loop, 2 turns, R = 1 m, I = 20 A,
CCW (from above)
Find: Torque (magnitude and direction)

13. Example 2
50o I
Which direction does the torque vector point in for a flat horizontal coin on the surface of earth ?

14. Example 3
Find the torque on a flat, horizontal, circular coil due to the magnetic field of the Earth.
50o
north
0.5 x 10-4 T
Circular loop, 2 turns, R = 1 m, I = 20 A (ccw from above)

15. Example 4
Consider an electron orbiting a proton and maintained in a fixed circular path of radius R=5.29x10-11 m by the Coulomb force.
Treating the orbiting charge as a current loop, calculate the resulting torque when the system is in an external magnetic field of B=0.4 T directed perpendicular to the magnetic moment of the orbiting electron.

16. Summary
Charged Particle:
Wire:
Straight wire, Uniform B: Torque: