Mathematical Tools for Physics

8—Multivariable Calculus 225

increasing coordinateφ. Finallyθˆis perpendicular to the coneθ=constant and again, points toward increasing
θ. Thenφˆ=ˆr×θˆ, and on the Earth, these vectorsˆr,ˆθ, andφˆareup,ˆ South, andˆ East.ˆ

Solenoid
A standard solenoid is cylindrical coil of wire, so that when the wire carries a current it produces a magnetic field.
To describe this field, it seems that cylindrical coordinates are advised. Until you know something about the field
the most general thing that you can write is

B~(r,θ,z) =ˆr Br(r,θ,z) +ˆθ Bθ(r,θ,z) +z Bˆ z(r,θ,z)

In a real solenoid that’s it; all three of these components are present. If you have an ideal, infinitely long solenoid
(found only in textbooks) the use of Maxwell’s equations and appropriately applied symmetry arguments will
simplify this toˆz Bz(r).

Gravitational Field
The gravitational field of the Earth is simple,~g=−r GM/rˆ^2 , pointing straight toward the center of the Earth.
Well no, not really. The Earth has a bulge at the equator; its equatorial diameter is about 43 km larger than its
polar diameter. This changes the~g-field so that it has a noticeableθˆcomponent. At least it’s noticeable if you’re
trying to place a satellite in orbit or to send a craft to another planet.
A better approximation to the gravitational field of the Earth is

~g=−ˆr

GM

r^2

−G

3 Q

r^4

[

ˆr

(

3 cos^2 θ− 1

)

/2 +ˆθcosθsinθ

]

(16)

The letterQstands for the quadrupole moment.|Q|MR^2 , and it’s a measure of the bulge. By convention a
football (American football) has a positiveQ; the Earth’sQis negative. (What about a European football?)

Nuclear Magnetic Field
The magnetic field from the nucleus of many atoms (even as simple an atom as hydrogen) is proportional to

1

r^3

[

2 ˆrcosθ+ˆθsinθ

]

(17)

As with the preceding example these are in spherical coordinates, and the component along theφˆdirection is
zero. This field’s effect on the electrons in the atom is small but detectable. The magnetic properties of the