A video of this demonstration is available at this link.

A coil of 4-1/2 turns of stiff wire is suspended between the poles of a magnet (north pole at left, labeled with a red N). The ends of this coil are connected through a tap switch and a reversing switch to a DC power supply. A current-limiting resistor prevents the power supply from being overloaded. With the reversing switch in one direction (toward the rear of the table, as in the photograph), the positive terminal of the power supply is connected to the top of the apparatus, and when you press the tap switch, current flows through the coil in the clockwise direction (viewing the apparatus as it appears in the photograph) and back to the power supply. When you throw the switch in the opposite direction (toward the front of the table, in the photograph), the positive terminal is connected to the bottom of the apparatus, and when you press the tap switch, current flows through the coil in the counterclockwise direction and back to the power supply. (The foregoing statements follow the convention that current refers to the flow of positive charge.) The current flowing through the coil experiences forces as it circulates within the field of the magnet. The current in the vertical sections of the coil experiences equal and opposite forces that point horizontally and thus exert a torque about the axis along which the coil is suspended (vide infra). With the reversing switch oriented as shown, this torque is clockwise (as viewed from the top), and when you press the tap switch, the coil and needle flip to the left. We can also think of the magnetic field generated by the current in the coil, which when the coil is situated as in the photograph, is at right angles to the field of the magnet. With the reversing switch oriented as shown, the current flows clockwise, and the north side of the magnetic field it produces faces the back of the table. The repulsion of like poles and the attraction of unlike poles of the two magnetic fields produces a clockwise torque on the coil so that when you press the tap switch, the coil and needle flip to the left. With the reversing switch toward the front of the table, when you press the tap switch, the coil and needle flip to the right.

As noted above, there are two ways in which we can examine the operation of this demonstration. Demonstrations 68.21 -- Jumping wire, 68.24 -- Current-carrying rod rolls into and out of ‘U’ magnet, and 68.27 -- Current-carrying aluminum foil under ‘U’ magnets, illustrate the force experienced by a current-carrying conductor in a magnetic field. (Demonstrations 68.39 -- Force between current-carrying wires, and 68.42 -- Collapsing coil, illustrate the force experienced by a current-carrying conductor in the magnetic field of a neighboring current-carrying conductor.) This force is the cross product of the current and the magnetic field, or F = il × B, where l is the length of the part of the conductor that is in the magnetic field. This is a consequence of the force felt by the individual charges, which for each charge is F = qv × B. If we consider electron flow, q = e, and v = v_d, the drift speed of the electron in the wire, and F_e = ev_dB sin θ. v_d = j/ne, where j is the current density (i/A,  where A is the cross-sectional area of the wire), and n is the number of conduction electrons per unit volume in the wire. Thus,  F_e = jB/n. The length of wire in the field, l, contains nAl conduction electrons, so the total force on the wire is F = (nAl)F_e = nAl(jB/n). Since jA = i, this gives the magnitude of the force as F = ilB sin θ, where θ is the angle between the direction of the current and the direction of the magnetic field. If the current is perpendicular to the field, F = ilB.

For the coil in this demonstration, the top and bottom (horizontal) sections probably sit outside the magnetic field, or at least in a region where the field is relatively weak. For this analysis, though, we will treat the coil as if all of it is in the magnetic field. Let us call the length of the top and bottom sections of the coil a. If current flows through the coil, the force on the top and bottom section of each turn is ia × B. With the coil oriented as in the photograph, these sections are parallel to the magnetic field, so the cross product is zero. If the coil were at some other angle with respect to the field, though, there would be equal and opposite vertical forces on the top and bottom sections of each turn. Since these forces are parallel to the axis along which the coil is suspended, they would not result in any net force on the coil. To connect the coil at top and bottom, there is a section that runs half the width of the loop. Since these are on the same side of the coil, the forces on them balance, and they do not introduce any net force on the coil.

If we call the length of the sides of the loops b, since these sections are perpendicular to the magnetic field, each experiences a force of ibB. If current flows from the top connection to the bottom, it flows upward on the left side of each loop, and downward on the right. The left side of each loop experiences a horizontal force directed toward the back of the table, and the right side of each loop experiences a force directed toward the front of the table. Each side of a single loop thus exerts a torque about the suspension axis of (ibB)(a/2) sin θ, where θ is the angle between the normal to the coil and the magnetic field. For a single turn, then, the torque is twice this, or (iabB) sin θ. The torque on a coil that has N turns, then, is τ = NiabB sin θ. Since our coil has four and a half turns, there are four vertical sections on the left and five on the right, so the torque is 4.5 (iabB) sin θ. This torque is clockwise when we view the coil from the top, and the needle flips to the left. If we reverse the current flow, the forces on the sides of the coil switch direction, the torque is counterclockwise, and the needle flips to the right. In the last equation, ab is just the area of the coil, which we can call A, and the torque is τ = NiAB sin θ. One can show that this equation holds for any plane loop of area A, whether it is rectangular or not.

When you remove the current from the coil, its suspension twists it so that the arrow (which is along its axis) points in front of the magnet’s axis, so you can switch the current back and forth, and the arrow will flip back and forth.

In the discussion above, we considered the forces that the different sections of the loops experience because of the current flowing through them as they sit in the magnetic field. We may also consider the magnetic field produced by the current flowing in the loop, and its interaction with the external magnetic field. As illustrated in demonstration 68.13 – Right-hand rule model, given a coil with current flowing in it, if you curl the fingers of your right hand in the direction of the current, your thumb points in the direction of the magnetic dipole, i.e., toward the north pole. By convention, current refers to the flow of positive charge, which is in the opposite direction to the flow of electrons.

In the first case, then (switch toward the rear), with a clockwise current flow, the north pole faces the rear of the table, and the south pole faces the front. The north pole of the magnet, labeled with the red “N,” repels the north pole of the coil’s magnetic field and attracts its south pole, twisting the coil clockwise (as viewed from the top) and flipping the red arrow to the left. (Of course, the south pole of the magnet repels the south pole of the coil’s magnetic field and attracts its north pole, which does the same thing.)

In the second case (switch toward the front), with a counterclockwise current flow, the north pole faces the front of the table, and the south pole faces the rear. As above, the north pole of the magnet repels the north pole of the coil’s magnetic field and attracts its south pole (and, as above, the magnet’s south pole repels the south pole of the coil’s magnetic field and attracts its north pole), but since the north pole is now in front (and the south pole is in back), this twists the coil counterclockwise (as viewed from the top), and the arrow flips to the right.

Again, when you remove the current from the coil, its suspension twists it so that the arrow (which is along its axis) points in front of the magnet’s axis, so you can switch the current back and forth, and the arrow will flip back and forth.

The magnetic dipole, μ, associated with an electric current flowing in a loop of wire is μ = NiA, where N is the number of turns in the loop, i is the current (in amperes) and A is the cross-sectional area of the loop. The area of the loop in this apparatus is approximately 3 cm × 5.5 cm = 16.5 cm, so for a given current, you can estimate the magnitude of the coil’s magnetic dipole. The torque on the coil, τ, is μ × B, where B is the field of the magnet. The magnitude of this torque depends on the orientation of the coil with respect to the magnetic field lines, and is τ = NiAB sin θ, or τ = μB sin θ, where θ is the angle between the magnetic field lines and the normal to the plane of the coil. We see that for the coil in this demonstration, the torque is maximum when the needle is pointing outward, as shown in the photograph, so that the normal to the plane of the coil is 90 degrees to the magnetic field lines. The torque is minimum (zero) when the coil and needle are turned 90 degrees to either side, and the normal to the plane of the coil is parallel to the magnetic field lines. As we should expect, we also see that the expression for the torque is the same as the one we obtained above.

If the coil were suspended by springs with known force constant, it would be possible, by measuring the displacement of the arrow for a given current flow, to determine the magnitude of the field of the magnet. (Conversely, of course, if we knew the magnetic field strength, we could determine the force constant of the springs.) This is the principle behind the d’Arsonval movement (also called “moving coil”), which is used in all analog meters (voltmeters, ammeters, etc.). In such a meter, a coil having a specific area and number of turns sits between the poles of a permanent magnet. The inner magnet pole piece is cylindrical, and the outer pole pieces are curved so that they provide a uniform radial magnetic field over the region in which the sides of the coil move. As a result, the magnetic field is always perpendicular to the normal to the plane of the coil, and the magnitude of the torque always equals NiAB (or μB). The coil is mounted on a pivot, and has attached to it an indicating needle and one or two hairsprings (spiral-shaped springs). The hairsprings balance the electromagnetic torque so that the deflection of the needle is proportional to the current flowing through the coil, and they return the needle to its zero position when the current stops flowing. The meter is designed so that a particular current gives the needle a full-scale deflection. A scale marked in divisions of that full-scale current allows direct reading of whatever the meter is measuring. To use such a meter as a voltmeter, one inserts a series resistor of the appropriate size to keep the reading on scale for a particular maximum voltage. In this arrangement, V_fs = I_fs(R_meter + R_series), where “fs” stands for “full-scale,” and I_fs = V_fs/(R_meter + R_series). The current, and thus the scale reading, is proportional to the voltage applied across the meter and series resistor. To use the meter as an ammeter (current meter), one places a shunt resistor of the appropriate size across the meter, forming a current divider. In this case, a predetermined portion of the measured current flows through the meter, and the rest flows through the shunt resistor. So I_meter/I_total = [R_shunt/(R_meter + R_shunt)]. One chooses a shunt for which, for a particular maximum current, the portion flowing through the meter gives a full-scale reading. (In the parenthetical reference above to types of meters, the “etc.” refers to applications in which some other quantity, for example, pressure or temperature, is converted to an electric current or voltage, and the scale is calibrated in the appropriate units for the quantity being measured.)

References:

1) Halliday, David and Resnick, Robert. Physics, Part Two, Third Edition (New York: John Wiley and Sons, 1978), pp. 758-62, 725-7.