Introductory Physics Volume Two

92 Magnetic Fields 4.6
The net force is
⃗F = îF 1 − ĵF 2
= îIL 1 B − ĵIL 2 B
So the direction is given by
( ) ( )
θ = tan −1 IL2 B
= tan −1 L2
,
IL 1 B
L 1
where θ is below the x-axis. The magnitude of the force is
√
F =
√F1 2 + F 2 2 = IB L 2 1 + L2 2
Example
What is the net force on an a × b rectangular loop of wire carrying a
current I that is in a uniform magnetic field perpendicular to the loop’s
plane?
Assume a direction for the magnetic field and use the right hand rule
to draw the forces on each segment of the loop:
a
F 1
b
F 2
F 3
F 4
B
The net force on the loop is:
⃗F = ⃗ F 1 + ⃗ F 2 + ⃗ F 3 + ⃗ F 4
= î(F 2 − F 4 ) + ĵ(F 3 − F 1 )
= î(IbB − IbB) + ĵ(IaB − IaB)
= 0
I
Example
Show that the magnetic force due to a uniform magnetic field for any
closed current loop is zero.
The force law states
∮
⃗F = I d ⃗ l × B ⃗