Introductory Physics Volume Two

4.7 Homework 93
Imagine the closed loop to be divided up into many d ⃗ l i , approximated
by a many-sided polygon:
B
I
So, the integral is just the limit of a sum over all of the small segments:
∮
d ⃗ l × ⃗ B = d ⃗ l 1 × ⃗ B + d ⃗ l 2 × ⃗ B + · · ·
= (d ⃗ l 1 + d ⃗ l 2 + · · ·) × B ⃗ (∮ )
= d ⃗ l × B ⃗
⃗B is outside the integral because it is constant. The vector sum indicated
by the integral in the last line must vanish since the vectors form
a close polygon; graphical vector addition yields zero. Thus,
∮
⃗F = I d ⃗ l × B ⃗
(∮ )
= I d ⃗ l × B ⃗
= 0
§ 4.7 Homework
⊲ Problem 4.7
Consider an electron near the equator. In which direction does it tend
to deflect by the magnetic field of the earth if its velocity is directed
(a) downward, (b) northward, (c) westward, or (d) southeastward?
⊲ Problem 4.8
An electron moving along the positive x axis perpendicular to a magnetic
field experiences a magnetic deflection in the negative y direction.
What is the direction of the magnetic field?
⊲ Problem 4.9
An electron in a uniform electric and magnetic field has a velocity of
1.2×10 4 m s in the positive x direction and an acceleration of 2.0×1012 m s 2