BACKGROUND AND THEORY

In this experiment we examine the measurement of an unknown magnetic field by two different methods. These methods involve the use of two laws of electromagnetism which you have already seen in your earlier work, namely Faraday's law of induction , and the force law for the interaction between a current and a magnetic field.

The appropriate laws will be restated for your use, and it will be assumed that you understood their derivation in your first year lecture course, or, in the unhappy event that this is not so, that you will consult a textbook for the missing details. There is considerable extra detail associated with the induction measurement; this also is an introduction to the use of an RC circuit as an integrator.

In MKS units, Faraday's law is:

V = - = -

where, you may recall, V is the integral around a closed contour, or in a practical sense, the voltage around a circuit, while is the total magnetic flux threading through the circuit. The minus sign came from Lenz's Law (and we'll drop it for the rest of this section). The units are:

V: Volts

: tesla m² (= webers)

The force on a current element dl due to a magnetic field B is:

= I

F: newtons

I: amperes

B: tesla

l: meters

Flip-Coil Technique

Having written down the essential mathematical relationships between magnetic field, potential, and current, we must now ask how we can use these to measure a constant magnetic field. Consider Faraday's law first. It relates a circuit voltage to the time derivative of magnetic flux though it; yet our problem is the measurement of a steady field which has no time variation at all. Furthermore, even if the field were varying, we are interested in its magnitude, not its rate of increase.

Let us arrive at a technique by considering the above two objections in reverse order. How do we get the magnitude of B (or ) from its derivative? By integrating it, obviously. But how does one integrate a voltage from an electrical circuit? There are several ways, actually, but the very simplest is one of the best. Imagine the following electrical network:

Figure 1 The RC integrator.

We will impress a time varying voltage V_i(t) at the input terminals at the left, and look at the resulting output voltage V_o(t) at the right.

Recall first that the voltage across a capacitor is proportional to the total charge Q placed on it; the constant of proportionality is the capacitance, C. Thus,

V = .

Since current, I, is a rate of charge transfer, it follows that:

Q(t₁) =

if Q = 0 at t = 0. Then,

V(t₁) = = V_o(t₁),

since our output voltage is just that across the capacitor. Now what is I? Ohm's law tells us that

I = ,

and so by direct substitution into Eq. 6,

V_o(t₁) =

= - .

If the product RC, which itself has the dimensions of a time, is much larger than the integration time t₁, the second right-hand term becomes small compared to V_o itself. (i.e., We approximate by V_ot₁, and so V_ot₁/RC <<V_o). Then,

V_o (t₁) = ,

our circuit acts an integrator.

So far, so good. We can connect such a integrator to a loop of wire in a magnetic field increasing from zero, and get an output V₀ which is related to the flux through the loop by (neglecting the negative sign, see Eq. 1)

V_o(t) = =

as long as RC>>t. If the loop has an area A, and the field strength B is uniform over this area, then = AB, and

V_o(t) = B(t)

Fine again. We can measure a magnetic field which has increased from zero in some suitable short time t. But how about measuring a steady field? You may already have anticipated the answer to this. We will simply take our loop which initially rests in a place where there is no field and thrust it suddenly into the magnetic region. By "suddenly", we mean in a time much shorter than RC. The loop itself then sees a sudden rising field, and develops an appropriate voltage at its terminals. In fact, it is just as easy, and in most cases easier, to have the coil at rest within the field, and then suddenly pull it out. The total change in B, which is the initial B itself, will appear as a change in V_o (= V_o) at the integrator output. Fig. 2 illustrates our scheme, which is called the flip-coil technique:

Figure 2

Sometimes the V_o we can get (while still satisfying the requirement that RC>>t₁) is inconveniently low. This is easily remedied by wrapping several turns, say n of them, on the loop. The output increases by a factor of n, since this is just equivalent to connecting n separate single-turn flip coils in series and thus adding their output voltages. In this case, then,

V_o = B

What happens, now, when the "flip" has occurred, and the integrator output voltage has jumped to V_o? We can answer this by taking another look at the circuit:

Figure 3 The RC integrator, discharging through the loop.

There is no more input voltage, since = 0 in the coil; the capacitor simply discharges through R and the coil, whose resistance is usually close to zero. You may recall that when a capacitor discharges through a resistance, its voltage decays as

V(t) = V_o

We conclude, then that the whole history of the integrator output voltage must look something like:

Figure 4

We can see now in what sense the flip of the coil must be sudden. If the discharge rate of the capacitor is anywhere near the charging rate during the flip of the coil, the output voltage will never reach it proper level. We must, then make t₁ much less than the discharge time constant which is RC; but this is simply restating our criterion for accurate integration by our integrator circuit.

There is one more experimental "fact of life" to take into account in this system. It is that the device which measures V_o(t) has something less than infinite resistance itself. The input resistance of the oscilloscope you will use is 10⁶ ohms, for example. The complete circuit for our measurement is then:

Figure 5 The flux integrator, with the oscilloscope input resistance shown as R₁.

It is clear that R and R₁ are in parallel as far as the discharge of the capacitor is concerned, and so the decay time constant is R₂C, where

R₂ = .

But what about the integration time constant? The answer is that this is still RC, and so

V_o = B,

where now the flip time t₁ must be kept small compared to R₂C. You should try to prove this to yourself.

Current-Balance Technique

The measurement of B by measuring the force on a current in the field is so straightforward as to hardly require an elaborate introduction at all. The only subtlety involved in such an experiment lies in the means by which one makes sure that the force which is measured is applied only to a well-defined segment of conductor in a homogeneous region of the field.

Suppose we were to try a measurement of the field between the poles of a magnet where the field has the following typical distribution:

Figure 6

We draw "lines" to represent B, as has become conventional, and use the further convention that their density corresponds to field strength. It is characteristic of this field distribution that the magnitude of B, i.e., the line density, becomes smaller in the "fringing" region, and goes gradually from its maximum value between the poles to zero far outside.

A wire carrying current I will then feel a total force

F_x = .

However, this doesn't tell the experimenter what B itself is at any point; we can only determine the integral B_z dy. If B were perfectly constant over the width of the pole faces and dropped abruptly to zero at the edges, then things would be simpler, and the integral would be B_zD, where D is the pole diameter; the field would then be given by

B_z = .

A strategy we might employ in this experiment would have I flow only in the homogeneous region of the field. A moment's thought exposes this as nonsense, however, since the current has to enter from the outside and return there again.

A more realistic and workable strategy would be to deploy the wire so that the forces experienced by the wire as it enters and leaves through the inhomogeneous part of the field (i.e., the fringing field cancel each other out, leaving only a net force from a section of the wire in the uniform field. How can these entering and leaving forces be made to cancel? Remember that the vector relation

= I

is resolved into

dF_x = I_yB_zdl

in the coordinates of our examples. Now, B_z does not change sign; therefore, if we want to generate two forces having opposite signs in the fringing field, we can only do it by using two oppositely directed currents. You should be ready by now to appreciate the unique properties of the following arrangement of wire in the region of the magnet poles. We are looking now along a line of B and toward the pole face:

Figure 7

The wire has the form of a hairpin, with three distinct straight segments, labeled 1, 2, 3, in the field. It should be clear that since I is equal but oppositely directed in legs 1 and 3, and since these pass through identical field distributions, their forces, F₁ and F₃ are also equal and opposite, and so add to zero.

All that remains is leg 2 of length l which is short enough to be everywhere in a uniform field. Therefore, the total force on the hairpin is

F₂ = IBl,

and one may now measure B by, for example, hanging the hairpin from a balance so as to measure F₂, and sending some carefully measured I through it. This is precisely what you will do in this experiment.