W9_parallel_resonance.eps

Week 7: Sources of the Magnetic Field 237

Example 7.4.2: Field of a Circular Loop on its Axis

dl

x

y

z

z r

dB

I

a

φ

φ

dBz

θ

dθ

dB

Figure 82: The geometry and coordinates used to compute the magnetic field of a circular loop wire
carrying a currentIon its axis of symmetry.

In the figure above, we have the geometry of a circular current loop in thexy-plane. Finding the
magnetic field of this loop at an arbitrary point in (say) spherical polar coordinates is not impossible,
but neither is it easy – it is a chore best left for your next course (if any) in electromagnetism. In
thiscourse, however, we can easily find the field at an arbitrary point onthez-axis, because there
we can use thecylindrical symmetryof the arrangement to our advantage.

We begin by writing the Biot-Savart Law for the small chunk of current in the segment of the
wire labelledd~l:

dB~=kmId

~l×rˆ

r^2

(514)

As you can see, the direction of this infinitesimal field element is in the plane formed by~rand the
z-axis, perpendicular to~rin the right-handed direction. The magnitude of this field element is:

dB=km

Idl

r^2

(515)

We can easily find the components ofdB~parallel and perpendicular tozusing the angleφ:

dBz = km

Idl

r^2 sin(φ) (516)

dB⊥ = km

Idl

r^2

cos(φ) (517)

We can evaluate the two trig functions using the right triangle with sides ofa,z, andr(which has
the same angleφin its apex) – sin(φ) =a/rand cos(φ) =z/r:

dBz = kmIdl a

r^3

(518)

dB⊥ = km

Idl z

r^3

(519)

At this point we could be lazy and invoke symmetry. The problem hasazimuthal symmetry– if
we walk around the ring and look at it from arbitrary angles, the problem does not change with our
perspective, so we know that thetotal magnetic field cannot have a component that changes as we
walk, that is, one in thexorydirection. The field can point in thezdirection only on thez-axis.
This allows us to evaluateBzonly to get the total field.