W9_parallel_resonance.eps

236 Week 7: Sources of the Magnetic Field

a general point on thex-axis, and no matter where we locate it we can do an integral betweenx 1
andx 2.

We therefore begin by considering the geometry of the cross product. If we let the fingers of our
right hand line up withdxin the direction ofIand rotate through the small angle to line up with
the vector~rbetweendxand the point of observation, our thumb picks the perpendicular direction
outof the page as indicated. We can thus easily write the magnitude of the magnitude for the field
produced by this small differential chunk of the current as:

dB=kmIsin(φ)dx

r^2

(507)

and hence (formally integrating both sides) we get:

B=

∫x 2

x 1

kmIsin(φ)dx

r^2

(508)

Alas, we have an embarrassment of variables in this. As we varyxin the integral, bothrand
φvary! To do the integral we have to useexactlythe same methodology we used to evaluate the
electricfield of a straight line ofcharge. We change variables, in other words, so that they are
consistently all (r, θ). That is:

x = ytan(θ)

dx =

y dθ

cos^2 (θ)

(509)

and
sin(φ) = cos(θ) (510)

(think about it for a minute, it will make sense). The integral becomes:

B=

∫θ 2

θ 1

kmIycos(θ)dθ

cos^2 (θ)r^2 =

∫θ 2

θ 1

kmIycos(θ)dθ

y^2 (511)

(usingy=rcos(θ)) and we finally obtain:

B =

kmI

y

∫θ 2

θ 1

cos(θ)dθ

= kmI

y

(sin(θ 2 )−sin(θ 1 )) (512)

(out of the page, as noted). This is almost exactly identical to our expression for the electric field
of a long straight line of charge as evaluated in week 2, so it should be easy enough to remember or
rederive.

As was the case then as well, we can find the magnetic field produced by aninfinitestraight
line of current by taking the limitsθ 1 → −π/2 andθ 2 →π/2, where the sines become−1 and 1
respectively. Note that this result will actually be relevant and useful any time we seek the field
close enoughto a wire carrying current that the angles to the end points approachπ/2. The field
of such an “infinite” wire is just:

B∞=

2 kmI

y

(513)

Again note the analogy with electric field, withke→kmandλ→I, but note well, thegeometryof
the field is entirely different! The magnetic field, for a finite or infinite wire carrying current, flows
incircular loops around the wire! In our picture, the field goes out of the page above the wire, into
the page below the wire, and in generalif we let the thumb of our right hand line up with
the direction of the current in the wire, the field circulatesaround the wire in the same
sense as our fingers.

This latter is, as we already saw in our original discussions of the magnetic field, a useful and
general rule as it will always work out to be the direction given by the cross product for a short
segment of current.