Scott Hughes
`
`10 March 2005
`
`Massachusetts Institute of Technology
`Department of Physics
`8.022 Spring 2004
`
`Lecture 10:
`Magnetic force; Magnetic fields; Ampere’s law
`
`10.1 The Lorentz force law
`
`Until now, we have been concerned with electrostatics — the forces generated by and acting
`upon charges at rest. We now begin to consider how things change when charges are in
`motion1.
`A simple apparatus demonstrates that something wierd happens when charges are in
`motion: If we run currents next to one another in parallel, we find that they are attracted
`when the currents run in the same direction; they are repulsed when the currents run in
`opposite directions. This is despite the fact the wires are completely neutral:
`if we put a
`stationary test charge near the wires, it feels no force.
`
`Figure 1: Left: parallel currents attract. Right: Anti-parallel currents repel.
`
`Furthermore, experiments show that the force is proportional to the currents — double the
`current in one of the wires, and you double the force. Double the current in both wires, and
`you quadruple the force.
`
`1We will deviate a bit from Purcell’s approach at this point. In particular, we will defer our discussion of
`special relativity til next lecture.
`
`89
`
`APPLE 1028
`
`

`

`This all indicates a force that is proportional to the velocity of a moving charge; and,
`that points in a direction perpendicular to the velocity. These conditions are screaming for
`a force that depends on a cross product.
`What we say is that some kind of field ~B — the “magnetic field” — arises from the
`current. (We’ll talk about this in detail very soon; for the time being, just accept this.) The
`direction of this field is kind of odd: it wraps around the current in a circular fashion, with
`a direction that is defined by the right-hand rule: We point our right thumb in the direction
`of the current, and our fingers curl in the same sense as the magnetic field.
`
`With this sense of the magnetic field defined, the force that arises when a charge moves
`through this field is given by
`
`~F = q
`
`~v
`c × ~B ,
`where c is the speed of light. The appearance of c in this force law is a hint that special
`relativity plays an important role in these discussions.
`If we have both electric and magnetic fields, the total force that acts on a charge is of
`course given by
`
`~F = q ~E +
`
`~v
`
`c × ~B! .
`
`This combined force law is known as the Lorentz force.
`
`10.1.1 Units
`
`The magnetic force law we’ve given is of course in cgs units, in keeping with Purcell’s system.
`The magnetic force equation itself takes a slightly different form in SI units: we do not include
`the factor of 1/c, instead writing the force
`
`~F = q~v × ~B .
`
`90
`
`

`

`This is a very important difference! It makes comparing magnetic effects between SI and cgs
`units slightly nasty.
`Notice that, in cgs units, the magnetic field has the same overall dimension as the electric
`field: ~v and c are in the same units, so ~B must be force/charge. For historical reasons, this
`combination is given a special name: 1 dyne/esu equals 1 Gauss (1 G) when the force in
`question is magnetic.
`(There is no special name for this combination when the force is
`electric.)
`In SI units, the magnetic field does not have the same dimension as the electric field: ~B
`must be force/(velocity × charge). The SI unit of magnetic field is called the Tesla (T): the
`Tesla equals a Newton/(coulomb × meter/sec).
`To convert: 1 T = 104 G.
`
`10.2 Consequences of magnetic force
`
`Suppose I shoot a charge into a region filled with a uniform magnetic field:
`
`B
`
`v
`
`The magnetic field ~B points out of the page; the velocity ~v initially points to the right. What
`motion results from the magnetic force?
`At every instant, the magnetic force points perpendicular to the charge’s velocity —
`exactly the force needed to cause circular motion. It is easy to find the radius of this motion:
`if the particle has charge q and mass m, then
`
`=
`
`Fmag = Fcentripetal
`mv2
`qvB
`c
`R
`mvc
`qB
`
`R =
`
`.
`
`91
`
`

`

`If the charge q is positive, the particle’s trajectory veers to the right, vice versa if its negative.
`This kind of qualitative behavior — bending the motion of charges along a curve — is typical
`of magnetic forces.
`Notice that magnetic forces do no work on moving charges:
`moves for a time dt, the work that is done is
`dW = ~F · d~s = ~F · ~v dt
`= q ~v
`c × ~B! · ~v dt
`
`if we imagine the charge
`
`= 0 .
`The zero follows from the fact that ~v × ~B is perpendicular to ~v.
`
`10.3 Force on a current
`
`Since a current consists of a stream of freely moving charges, a magnetic field will exert a
`force upon any flowing current. We can work out this force from the general magnetic force
`law.
`Consider a current I that flows down a wire. This current consists of some linear density
`of freely flowing charges, λ, moving with velocity ~v. (The direction of the charges’ motion
`is defined by the wire: they are constrained by the wire’s geometry to flow in the direction
`it points.) Look at a little differential length dl of this wire (a vector, since the wire defines
`the direction of current flow).
`The amount of charge contained in this differential length is dq = λ dl. The differential
`of force exerted on this piece of the wire is then
`
`~v
`c × ~B .
`There are two equivalent ways to rewrite this in terms of the current. First, because the
`current is effectively a vector by virtue of the velocity of its constituent charges, we put
`~I = λ~v and find
`
`d ~F = (λ dl)
`
`~I
`c × ~B .
`Second, we can take the current to be a scalar, and use the geometry of the wire to define
`the vector:
`
`d ~F = dl
`
`I c
`
`d~l × ~B .
`These two formulas are completely equivalent to one another. Let’s focus on the second
`version. The total force is given by integrating:
`
`d ~F =
`
`Z d~l × ~B .
`
`I c
`
`~F =
`
`If we have a long, straight wire whose length is L and is oriented in the ˆn direction, we find
`
`IL
`ˆn × ~B .
`c
`This formula is often written in terms of the force per unit length: ~F /L = (I/c)ˆn × ~B.
`92
`
`~F =
`
`

`

`10.4 Ampere’s law
`
`We’ve talked about the force that a magnetic field exerts on charges and current; but, we
`have not yet said anything about where this field comes from. I will now give, without any
`proof or motivation, a few key results that allow us to determine the magnetic field in many
`situations.
`The main result we need is Ampere’s law:
`
`4π
`c
`
`Iencl .
`
`IC
`
`~B · d~s =
`In words, if we take the line integral of the magnetic field around a closed path, it equals
`4π/c times the current enclosed by the path.
`Ampere’s law plays a role for magnetic fields that is similar to that played by Gauss’s law
`for electric fields. In particular, we can use it to calculate the magnetic field in situations that
`are sufficiently symmetric. An important example is the magnetic field of a long, straight
`wire: In this situation, the magnetic field must be constant on any circular path around the
`
`wire. The amount of current enclosed by this path is just I, the current flowing in the wire:
`
`I ~B · d~s = B(r)2πr =
`
`4π
`c
`
`I
`
`2I
`cr
`
`→ B(r) =
`The magnetic field from a current thus falls off as 1/r. Recall that we saw a similar 1/r law
`not so long ago — the electric field of a long line charge also falls off as 1/r. As we’ll see
`fairly soon, this is not a coincidence.
`The direction of this field is in a “circulational” sense — the ~B field winds around the
`wire according to the right-hand rule2. This direction is often written ˆφ, the direction of
`
`.
`
`2In principle, we could have defined it using a “left-hand rule”. This would give a fully consistent
`description of physics provided we switched the order of all cross products (which is identical to switching
`the sign of all cross products).
`
`93
`
`

`

`increasing polar angle φ. The full vector magnetic field is thus written
`
`~B =
`
`2I
`cr
`
`ˆφ .
`
`10.4.1 Field of a plane of current
`
`The magnetic field of the long wire can be used to derive one more important result. Suppose
`we take a whole bunch of wires and lay them next to each other:
`
`θ
`
`y
`
`L
`
`x
`
`The current in each wire is taken to go into the page. Suppose that the total amount of
`current flowing in all of the wires is I, so that the current per unit length is K = I/L. What
`is the magnetic field at a distance y above the center of the plane?
`This is fairly simple to work out using superposition. First, from the symmetry, you
`should be able to see that only the horizontal component of the magnetic field (pointing
`to the right) will survive. For the vertical components, there will be equal and opposite
`contributions from wires left and right of the center. To sum what’s left, we set up an
`integral:
`
`ˆxZ L/2
`
`−L/2
`
`2 c
`
`~B =
`
`(I/L)dx
`√x2 + y2
`In the numerator under the integral, we are using (I/L)dx as the current carried by a “wire”
`of width dx. Doing the trigonometry, we replace cos θ with something a little more useful:
`
`cos θ .
`
`y
`√x2 + y2
`
`−L/2
`2Ky
`c
`
`−L/2
`
`(I/L)dx
`ˆxZ L/2
`√x2 + y2
`dx
`ˆxZ L/2
`x2 + y2 .
`This integral is doable, but not particularly pretty (you end up with a mess involving arc-
`tangents). A more tractable form is obtaining by taking the limit of L → ∞: using
`dx
`π
`Z ∞
`,
`x2 + y2 =
`|y|
`
`2 c
`
`~B =
`
`=
`
`−∞
`
`we find
`
`~B = ˆx
`
`2πKy
`c|y|
`
`94
`
`

`

`= +ˆx
`
`y > 0 (above the plane)
`
`2πK
`c
`2πK
`c
`
`= −ˆx
`The most important thing to note here is the change as we cross the sheet of current:
`
`y < 0 (below the plane)
`
`4πK
`c
`
`.
`
`|∆ ~B| =
`Does this remind you of anything? It should! When we cross a sheet of charge we have
`|∆ ~E| = 4πσ .
`The sheet of current plays a role in magnetic fields very similar to that played by the sheet
`of charge for electric fields.
`
`10.4.2 Force between two wires
`
`Combining the result for the magnetic field from a wire with current I1 with the force per
`unit length upon a long wire with current I2 tells us the force per unit length that arises
`between two wires:
`
`| ~F|
`L
`
`=
`
`2I1I2
`c2r
`
`.
`
`Using right-hand rule, you should be able to convince yourself quite easily that this force is
`attractive when the currents flow in the same direction, and is repulsive when they flow in
`opposite directions.
`
`10.4.3 SI units
`
`In SI units, Ampere’s law takes the form
`
`IC
`~B · d~S = µ0Iencl
`where the constant µ0 = 4π × 10−7 Newtons/amp2 is called the “magnetic permeability of
`free space”. To convert any cgs formula for magnetic field to SI, multiply by µ0 × (c/4π).
`For example, the magnetic field of a wire becomes
`
`~B =
`
`µ0I
`2πr
`
`ˆφ .
`
`The force between two wires becomes
`| ~F|
`L
`
`=
`
`µ0I1I2
`2πr
`
`.
`
`If you try to reproduce this force formula, remember that the magnetic force in SI units does
`not have the factor 1/c.
`
`95
`
`

`

`10.5 Divergence of the ~B field
`
`Let’s take the divergence of straight wire’s magnetic field using Cartesian coordinates. We
`put r = √x2 + y2. With a little trigonometry, you should be able to convince yourself that
`ˆφ = ˆy cos φ − ˆx sin φ
`xˆy
`yˆx
`√x2 + y2 −
`√x2 + y2
`=
`
`,
`
`so
`
`~B =
`
`2I
`
`c " xˆy
`x2 + y2 −
`
`yˆx
`
`x2 + y2# .
`
`The divergence of this field is
`
`2I
`
`c "
`
`~∇ · ~B =
`= 0 .
`
`2yx
`(x2 + y2)2 −
`
`2xy
`
`(x2 + y2)2#
`
`We could have guessed this without doing any calculation: if we make any small box, there
`will be just as many field lines entering it as leaving.
`Although we have only done this in detail for this very special case, it turns out this
`result holds for magnetic fields in general:
`
`~∇ · ~B = 0
`Recall that we found ~∇ · ~E = 4πρ — the divergence of the electric field told us about the
`density of electric charge. The result ~∇ · ~B = 0 thus tells us that there is no such thing as
`magnetic charge.
`This is actually not a foregone conclusion: there are reasons to believe that very small
`amounts of magnetic charge may exist in the universe, created by processes in the big bang.
`If any such charge exists, it would create a “Coulomb-like” magnetic field, with a form just
`like the electric field of a point charge. No conclusive evidence for these monopoles has ever
`been found; but, absence of evidence is not evidence of absence.
`
`10.6 What is the magnetic field???
`
`This “magnetic field” is, so far, just a construct that may seem like I’ve pulled out of the air.
`I haven’t pulled it out of the air just for kicks — observations and measurements demonstrate
`that there is an additional field that only acts on moving charges. But what exactly is this
`field? Why should there exist some field that only acts on moving charges?
`The answer is to be found in special relativity. The defining postulate of special relativity
`essentially tells us that physics must be consistent in every “frame of reference”. Frames of
`reference are defined by observers moving, with respect to each other, at different velocities.
`Consider, for example, a long wire in some laboratory that carries a current I. In this
`“lab frame”, the wire generates a magnetic field. Suppose that a charge moves with velocity
`~v parallel to this wire. The magnetic field of the wire leads to an attractive force between
`the charge and the wire.
`
`96
`
`

`

`Suppose we now examine this situation from the point of view of the charge (the “charge
`frame”). From the charge’s point of view, it is sitting perfectly still. If it is sitting still, there
`can be no magnetic force! We appear to have a problem: in the “lab frame”, there is
`an attractive magnetic force. In the “charge frame”, there can’t possibly be an attractive
`magnetic force. But for physics to be consistent in both frames of reference, there must be
`some attractive force in the charge frame. What is it???
`There’s only thing it can be:
`in the charge’s frame of reference, there must be an
`attractive ELECTRIC field. In other words, what looks like a pure magnetic field in one
`frame of reference looks (at least in part) like an electric field in another frame of reference.
`To understand how this happens, we must begin to understand special relativity. This is our
`next topic.
`
`97
`
`