W9_parallel_resonance.eps

214 Week 6: Moving Charges and Magnetic Force

The forces between the ∆xparts of the curve (the direction that would have been into or out of
the page in the rectangular figure above) arealsoequal and opposite, but they are typically offset
so that they do not act along a common line but rather one with a perpendicular displacement of
ysin(θ), whereθis the angle between theB~field and a right handed normal to the figure.y(for this
small segment of current) thus acts like theacoordinate in the rectangular figure above, ∆xacts
like a very short piece of thebsegment. This pair of forcesdoescontribute a net torque (magnitude)
for just this little strip of the total wire of:

∆τ=y∆xN IBsin(θ) (457)

Summing over all of the strips of width ∆x, the total torque on this plane loop is thus:

τ=N I lim

∆x→ 0

(∑

y(x)∆x

)

Bsin(θ) =N I

(∫

y(x)dx

)

Bsin(θ) =N IABsin(θ) (458)

or (including the vector direction from the right-hand-rule applied both to the torque and the right
handed normal to the loop):
~τ=~m×B~ (459)

with
~m=N IAnˆ (460)

We see that our rule for the rectangular loop above is thusgeneraland applies to any plane loop of
current, no matter what the shape.

0.1 Potential Energy of a Magnetic Dipole

As before with electric dipoles, we must doworkrotating a magnetic moment from one angle to
another in a magnetic field, working against the torque. The workwedo to rotate the dipole equals
the potential energy stored in the system (the magnetic dipole andfield combined). We can compute
this potential energy by following the derivation we used for electricdipoles, using as before a zero
of the potential energy when the dipole is at right-angles to the magnetic field. That is (given
τ=−mBsin(θ), with sign opposite to the sign ofθ):

U = −

∫

τ dθ

= −

∫θ

π/ 2

(−mBsin(θ))dθ

= −mBcos(θ)

or
U=−~m·B~ (461)

Note that as before,U(θ) is minimum (negative) when the magnetic dipole is aligned with the field,
maximum (positive) when antialigned.

From this, we can also find the force acting on a magnetic dipole in anon-uniform magnetic
field:

Fx=−

dU

dx (462)

(with similar expressions for the other force components, where this derivative should really be a
partial derivativefor those of you who have taken multivariate calculus).

We can now construct a table of the analogies between electric and magnetic dipole moments
and their associated fields, forces, and torgues. It is quite strong: