Force on a current-carrying wire

What you are trying to do

Measure the force on a current-carrying wire in a magnetic field with a top-pan balance, and test F = BIl.

Apparatus

• A pair of magnadur magnets on an iron yoke, standing on a top-pan balance reading to 0.01 g
• A stiff copper wire clamped rigidly so it passes horizontally between the poles without touching them
• DC supply, ammeter and rheostat; a switch
• A rule to measure the length of the magnet region

Variables

• Independent: the current in the wire
• Dependent: the force, read as the change in the balance reading
• Control: the field (same magnets), the length in the field, and the geometry: wire horizontal and at right angles to the field

Method

1. Zero (tare) the balance with the wire in place and the current off.
2. Switch on, read the change in the balance reading, switch off. Repeat at a series of currents, and reverse the current once to see the reading change sign.

1. Measure the magnet length and take it as an approximation to the effective length of wire in the field: the field falls away quickly, though not sharply, beyond the poles.

Analysis

1. The balance reads mass, so convert: F = Δm × g, with Δm in kilograms.
2. Plot F against I. The line through the origin has gradient Bl, so the flux density is the gradient divided by the measured length.
3. What one straight line tests. F against I through the origin, with a gradient giving a flux density of the right size for magnadur, supports the current part of F = BIl. It leaves the length and the angle untested, because you held both fixed all afternoon. If you want the rest of the equation, thread the wire through the gap two and then three times so the same current crosses the field twice and three times over, and check that the force follows. That is a different experiment in the same clamp stand, and it is the one the equation deserves.
4. What the balance is really weighing. It reads a vertical force and cannot tell a magnetic one from a stiff lead pressing on the yoke or a draught across the pan. It also reports a force spread over a length that is not sharply defined, since the field fringes past the ends of the magnets, so the l you divide by is a little shorter than the length the current actually feels, and B comes out a little high. Repeating readings does nothing for either. A Hall probe in the gap gives an independent B, which is the check that settles it.

A worked set of readings

Magnadur pair of length 5.0 cm; balance tared at zero current:

Idealised illustrative data, chosen so the working is easy to follow. Real readings scatter about the line rather than sitting on it, and your own graph will have points either side of the best fit.

The gradient of F against I is 4.0 mN A⁻¹, which is Bl. Dividing by l = 0.050 m gives B = 0.080 T, a typical magnadur field.

Evaluating the result

Idealised readings with one taken while a lead rested on the yoke. Changing the current means moving leads, and a stiff wire leaning on the magnet assembly presses on the balance exactly as a magnetic force would. The third column divides F by I, which the model says is Bl and the same at every current.

Idealised illustrative data, chosen so the working is easy to follow. Real readings scatter about the line rather than sitting on it, and your own graph will have points either side of the best fit.

Three rows return 4.00 mN A⁻¹ and the disturbed one returns 4.30. The test that settles it is built into the apparatus: reverse the current. A magnetic force changes sign, so the balance reading should swing the other way by the same amount, while a lead resting on the yoke pushes the same way whichever way the current runs. If the two readings are not equal and opposite, the difference is mechanical, and the cure is to dress the leads so they hang free rather than to average harder.

Where the uncertainty comes from

• The field's edges: The field is not a sharp box: it fringes beyond the magnets, so the effective length is slightly more than the magnet length. This is usually the dominant systematic.
• Balance resolution: 0.01 g is a tenth of a millinewton; small currents produce changes only a few times that, so the graph leans on the larger currents.
• Geometry: The force falls with the sine of the angle between wire and field; clamp the wire square and horizontal.
• Heating: Amps through a thin wire warm it; switch off between readings.

What earns the marks

Safety