Introductory Physics Volume Two

4.3 Trajectories Under Magnetic Forces 83
y
dl
R
dθ
θ
x
There is a uniform magnetic field in the y direction, B ⃗ = Bĵ. We
wish to compute the net force on this section of wire without using the
theorem F ⃗ = I∆l ⃗ × B. ⃗
(a) If we break the semicircle into small sections, they will be small
sections of arc, as pictured in the diagram above. If we take one of
these, it will be at a position θ, and will subtend an angle dθ. Show
that dl ⃗ = (− sin θî + cos θĵ)Rdθ
(b) With this result you can now compute the integral F ⃗ = ∫ I dl ⃗ × B. ⃗
Show that the net force on the semicircle is −I2RBˆk.
(c) Show that this is the same answer you get when you apply the
theorem F ⃗ = I∆l ⃗ × B. ⃗
⊲ Problem 4.2
Consider the rectangular current loop pictured below.
w
I
B
B
The field is uniform and in the direction indicated in the diagram.
Show that the torque about the axis indicated by the dotted line is
IAB where A is the area of the loop.
§ 4.3 Trajectories Under Magnetic Forces
Because the magnetic force is perpendicular to the velocity of the
particle, the magnetic force can not do work on the particle. Suppose
that a particle moves a small distance ⃗ dr. The work done by a force ⃗ F
as the particle moves is
dW = ⃗ F · ⃗dr
But ⃗ dr = ⃗v dt so
dW = ⃗ F · ⃗v dt
h