Slowly Varying Fields in Vacuum
3.1 Magnetic Induction
Chapter 3
The name is rather a misnomer because the EMF is certainly not a force, rather, it is the work that would be performed on a unit charge in travelling around the loop. In electrostatics it makes no difference whether forces arising from the static electric fieldE are included in F·d, because this contribution would sum to zero in any case.
To clarify how chemistry might give rise to nonelectrostatic forces, we digress briefly to the specific example of a dilute solution of an electrolyte such as HCl, whose concentration varies spatially. The electrolyte will be almost entirely disso- ciated into H+ and Cl− ions. H+, being much lighter, diffuses more rapidly than Cl−; therefore, more H+ ions than Cl− diffuse into regions of low concentration.
If the concentration gradient is maintained, a net positive current will flow into the low-concentration region until the accumulation of excess charge produces an electric field large enough to counter the differential diffusion. We might usefully think of the diffusion resulting from a forceF causing the movement of the ions. In terms of this force, the equilibrium condition becomes F +e E= 0. Clearly, inside the medium, e E=−F. We could build a battery on this principle–separating two halves of a container with a permeable membrane, filling one side of the container with HCl and filling the other with clear water.
3.1.2 Magnetically Induced Motional EMF
When a charge is forced to move through a magnetic induction field, it is subjected to a force (1–30) due to motion through the field:
F =qv×B (3–2)
Although the motion produced by this force gives the charged particle the ca- pacity to do work, it is important to recognize that the magnetic field does not do any work on the charge; instead, whatever agent produces or maintainsv does the work. Let us consider the EMF for a mobile loop placed in a static electric and magnetic field. In particular, we allow the loop to stretch and deform with velocity v(r, t). The force on a charge attached to the moving loop is then q(E +v×B),
Figure 3.1:A small segment of the loop increases the area at a ratev×d .
leading to an EMF
E=
Γ(t)
E ·d+
Γ(t)
(v×B) ·d (3–3)
For static fields, the first integral vanishes, and the triple product in the second integral may be rearranged to give
E=
Γ(t)
(d×v)·B (3–4)
During a timedt, a segment of the loop of length dmoves to increase the area within the loop by dS =vdt×d (Figure 3.1). Thus we write d×v =−dS/dt, and, instead of summing over the length of the loop, we sum the area increments:
E=−
Σ(t)
dS dt ·B where Σ(t) is the area included in the loop.
Defining the magneticflux, Φ, as Φ≡
Σ
B ·dS (3–6)
we see that in the case of static fields,
E =−dΦ
dt (3–7)
(It should, incidently, now be clear why B is sometimes called the magnetic flux density.)
Example 3.1: A flat circular coil ofN turns and radiusatravels in a direction parallel to the plane of the coil through a uniform magnetic induction field perpendicular to the plane of the coil. Find the EMF generated.
Solution: The flux through the loop is constant; hence, no EMF will be generated.
The EMF generated at the leading edge (semicircle) is precisely cancelled by the EMF at the trailing edge.
Example 3.2: A nova sheds a ring of ionized gas expanding radially through a uniform magnetic induction fieldBzkˆ with velocityv. Find the tangential acceleration of the charged particles in the ring.
Solution: The EMF generated around a loop of radiusr is F
q ·d=−dΦ
dt =−2πrBz
dr
dt (Ex 3.2.1)
The line integral of the tangential force is just 2πrFϕ, givingFϕ=qvrBz (a result we might have anticipated) and finally
aϕ=−q
mvBz (Ex 3.2.2)
(3–5)
negative charges are accelerated in the ˆϕdirection whereas positive charges are ac- celerated in the opposite direction. This conclusion could, of course, have been reached much more easily from a direct application of the Lorentz force F = q(dr/dt)×B.
3.1.3 Time-Dependent Magnetic Fields
Figure 3.2: The rectangular loop moves in thex direction through a non- homogeneous magnetic induction field.
Let us now consider a loop whose area does not change but instead moves through a magnetic induction field whose strength varies with position. The sides of the moving loop will evidently experience a time-dependent field. To simplify matters, consider a small rectangular loop of dimensionsδxandδyin thex-yplane, moving in thex direction through a magnetic induction field whosez component varies (to first order) linearly with x (Figure 3.2). The EMF generated around the moving loop is generally
E = F
q ·d= (v×B) ·d (3–8)
If the field is not homogeneous, having valueBz(x, y) atx andBz(x+δx, y) = Bz(x, y) + (∂Bz/∂x)δxon the other side of the loop, we expand the integral as
E=
δy
v×B(x, y )
· −ˆ dy+
δy
v×B(x +δx, y)
·ˆ dy (3–9)
The integrals over the sides parallel to the velocity make no contribution to the EMF and have therefore been neglected. Gathering the two terms, we have
E=
δy
v×∂ B(x, y)
∂x δx· dyˆ (3–10)
which becomes, on putting in the explicit directions ofvandB, E =
δy
ˆıdx
dt ×kˆ∂Bz(x, y)
∂x δx· dyˆ
=−
δy
∂Bz
∂t δx dy (3–11)
We conclude that
E=− ∂Bz
∂t dSz (3–12)
It is not difficult to generalize this result to fields and motions in arbitrary directions, to obtain for a loop of arbitrary, but constant, area
E =− ∂ B
∂t ·dS=−d
dt B ·dS (3–13)
We note that this result can again, as in (3–7), be written E=−d
dtΦ
If instead of moving the loop we move the magnet responsible for the field above, special relativity would require the same EMF, but since now the velocity v = 0, implying there can be no contribution fromv×B. Charges within the wire of the loop have no way to tell whether the loop is moving or some other means is used to vary the field temporally. The conclusion must then be that the first integral of (3–3) cannot vanish when we have temporally varying fields. Instead, we must have
E = E ·d=−dΦ
dt (3–14)
We note that as a consequence of (3–14), whenB varies in time, we cannot maintain a vanishing curl ofE.
Example 3.3: An electron with speed v executes cyclotron motion between the parallel faces of an electromagnet whose field is increased at a rate of dBz/dt.
Determine the tangential acceleration of the electron.
Figure 3.3:When the cyclotron fieldB and its derivatived B/dtare parallel, the charged particle increases its speed.
Solution: The radius of the electron’s orbit (Figure 3.3) is readily obtained from
−evBz=mv2
r ⇒ r= mv
−eBz
(Ex 3.3.1)
The flux included in the orbit is just Φ =πr2Bz, so that the EMF is given by E ·d= 2πrEϕ=−πr2dBz
dt (Ex 3.3.2)
giving rise to a tangential acceleration a= e E
m =− er 2m
dBz
dt ϕˆ= v 2Bz
dBz
dt ϕˆ (Ex 3.3.3)
The electron is accelerated in the direction of its motion.
3.1.4 Faraday’s Law
Moving the magnet is, of course, just one means of changing B within the loop as a function of time. We might equally well decrease or increase the current to an electromagnet or use other means of changing the field. We postulate that, under all conditions,
Γ
E·d=− d dt
Σ
B ·dS (3–15)
The relation (3–15) is Faraday’s law in integral form. We can obtain the differential form by applying Stokes’ theorem to the leftmost integral:
Σ
(∇ × E) ·dS=− d dt
Σ
B ·dS=−
Σ
∂ B
∂t ·dS (3–16)
Since Σ was arbitrary, the integrands must be equal, giving
∇ × E =−∂ B
∂t (3–17)
for the required result. We note that because∇ · B vanishes,∇ · (∇ × E) vanishes, as of course it must.