b)
Calculate the force on each side in x- and y- components.
c)
Add the components from all sides and show that the net force on the triangle is zero

P118 2018W WS09
Torque on a current-carrying loop
Consider the forces on the different sides of a current loop. The plane of the loop is at an angle
with respect to the
vertical axis.
We can calculate direction and
magnitude of the magnetic force
acting on each side using
?
𝐵
= 𝐼𝐿
⃗
× ?
⃗
The directions are given by the
right-hand rule and we find that
the forces on opposite sides are
of the same magnitude and point
in opposite directions.
The forces on the longer sides
are exerted around a moment
arm leading to a net torque. The
forces on the smaller sides are
along the same line and cancel.
The picture shows these forces,
viewed from the side as indicated
by the pair of eyes.
Q11.17
Show that the net torque on the rectangular loop can be written as
𝝉
⃗ = 𝝁
⃗ × 𝑩
⃗⃗
The
magnetic dipole moment
is defined as
𝜇 = 𝐼?
where
?
is the area vector of the loop and the direction is determined by a right-
hand rule.
I
is the current in the loop.
Start with the definition of the torque
𝜏 = ? × ?
and its magnitude
𝜏 = ?? sin𝜃
and the expressions for the torque:

P118 2018W WS09
Q11.18
If released from rest, the current loop will
A.
Move upward.
B.
Move downward.
C.
Rotate clockwise.
D.
Rotate counterclockwise.
E.
Do something not listed here.
Draw the directions of the forces on opposite sides of the ring.
(The picture shows a cut through the ring.)
Q11.19
A current loop, carrying a current of 3.5 A, is in the shape of an equilateral triangle with sides 20 cm. The loop
is in a uniform magnetic field of magnitude 100 mT whose direction is parallel to the current of the right side of the
loop.
a)
Find the magnitude of the magnetic dipole moment.
b)
Calculate the torque on the loop.
c)
Assuming that the loop is free to rotate in any direction, in which direction will it rotate? Draw the rotation axis in
the figure and show the torque as a vector.
Answers: 0.07 A m
2
; 0.007 N m

P118 2018W WS09
Potential Energy of a loop in a uniform magnetic field.
The
magnetic dipole moment tends to align with the external magnetic field
. This applies not only to loops, but also to
compass needles and atoms. For example, the magnetic interaction between electrons inside an atom gives rise to spin-
orbit and spin-spin interaction that influence their potential energy.
Q11.20
In the semi-classical Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius 5.3
x 10
-11
m with a speed of 2.2 x 10
6
m/s.
a)
What is the magnetic moment due to the moving electron?
b)
The potential energy of an object with magnetic dipole moment u is given by
𝑈 = −𝜇 ∙ ?
⃗
According to the Bohr model and assuming that a hydrogen atom is initially aligned in direction of a magnetic field
of strength 1.0 T, what energy would be needed to flip a hydrogen atom in opposite direction?

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- Fall '19