APhO 2009, Експеримент — Задача 1. The Earth’s Horizontal Magnetic Field

Experimental competition, 28 April 2009

Условие

This is to determine the horizontal component of the Earth’s magnetic field BHB_{\mathrm{H}} using small-amplitude oscillation of a cylindrical bar magnet. The magnet is to oscillate in the combined static fields of the Earth and that due to a square coil.

Perspective sketch with axes X, Y, Z meeting at O: a square coil of side a (labels a on its sides) in the YZ plane centred at O, with current leads marked i; along the X axis at distance x from O hangs a cylindrical magnet at point P on a supporting string, with a rotation arrow; labels 'square coil of sides a and N turns' and 'magnet of length L and radius r'.
Figure 1

The experiment is to be done in three sections. Section I is a derivation of formulae to be used in Section III.

Apparatus

Each student is provided with apparatus as shown in Figure 2:

  1. a square coil of resistance 5.2±0.2 Ω5.2 \pm 0.2\ \Omega and 130 turns
  2. a small cylindrical magnet of mass 15.0±0.215.0 \pm 0.2 g with nylon strings.
  3. a voltmeter (for measuring the potential difference across the coil only)
  4. a power supply (placed under the table to avoid the interference of its magnetic field)
  5. a wooden stand
  6. a stop watch
  7. a ruler
  8. a protractor
  9. white label (you can write on it)
  10. color clay
  11. graph papers
  12. an electrical cord
Three photographs of the apparatus with numbered labels 1–12: the wooden stand with the square coil (1), magnet (2), multimeter (3), stand (5), stop watch (6), ruler (7), protractor (8), clay (10), cord (12) and the platform; the power supply (4) under the table connected by the cord (12); a close-up of the coil terminals with the white label (9), terminals marked 'not used', and the note 'DO NOT change these connections.'
Figure 2

Warning

Use the multi-meter to measure only the voltage difference of the coils. Using the multi-meter in other modes can destroy the power supply!!!

Section I (1 т.)

It is given here that the magnetic flux density BPB_{\mathrm{P}} at a perpendicular distance ℓ\ell from the middle of a straight current element iaia is

BP=μ0i2πℓ (a/2)ℓ2+(a2)2(i)B_{\mathrm{P}} = \frac{\mu_0 i}{2\pi\ell}\,\frac{(a/2)}{\sqrt{\ell^2+\left(\frac{a}{2}\right)^2}} \qquad (i)

where μ0=4π×10−7\mu_0 = 4\pi\times10^{-7} henry per metre, the permeability of free space.

Use this expression to show that the expression for the magnitude of the magnetic flux density from the square coil at point P in Figure 1 is given by

Bpx=(μ0a2iN2π)[1(x2+(a2)2)x2+2(a2)2](ii)B_{px} = \left(\frac{\mu_0 a^2 iN}{2\pi}\right)\left[\frac{1}{\left(x^2+\left(\frac{a}{2}\right)^2\right)\sqrt{x^2+2\left(\frac{a}{2}\right)^2}}\right] \qquad (ii)

It is also given here that the period of a small-amplitude oscillation of the magnet in the net magnetic field BB is

T=2πImB(iii)T = 2\pi\sqrt{\frac{I}{mB}} \qquad (iii)

where mm is the magnetic moment of magnet with mass MM, and II is its moment of inertia about the axis through its centre of mass

I=M(L212+r24)(iv)I = M\left(\frac{L^2}{12}+\frac{r^2}{4}\right) \qquad (iv)
A straight current element of length a (double-headed arrow labelled a) carrying current i (arrow); from its middle a dotted perpendicular of length ℓ rises to point P, with a right-angle mark.
Figure 3

Section II (0,8 т.)

For the experiments in Section III you have to align the magnet in the position as shown in Figure 1. If the length of the string is too small, the torsion of the string cannot be neglected in the oscillation of the magnet. Perform appropriate measurements (say, oscillation of magnet in Earth’s magnetic field alone) to justify that we can ignore the torsion of the string. You are not required to plot a graph.

Section III

For the following experiments (in a, b, and c), you have to align the magnet in the position as shown in Figure 1. Measure and write down the value of the distance between the centre of the magnet and the top surface of the platform. [0.2 point]

a) Coil’s magnetic field and Earth’s horizontal magnetic field in the same direction

Warning

Please connect the coil to the power supply and leave it on for at least 5 minutes.

Measure periods of oscillation for different values of the combined field strength when the coil’s magnetic field and Earth’s magnetic field are in the same direction. Draw a straight line graph and compute the values of BHB_{\mathrm{H}} and the magnetic moment mm from this graph and estimate their errors. [5 т.]

b) Earth’s magnetic field only

Use the value of mm from (a) and the period of oscillation of the magnet bar in the absence of the Coil’s magnetic field from Section II to calculate again the value for BHB_{\mathrm{H}} and estimate its error. [1 т.]

c) Coil’s magnetic field and Earth’s horizontal magnetic field in opposite directions

By reversing the connection at the power supply, find the equilibrium position x0x_0 along the X-direction between Earth’s magnetic field and the opposing magnetic field from the coil. Use the value of x0x_0 to calculate again the value for BHB_{\mathrm{H}} and estimate its error. [2 т.]

Решение

Внимание: Непълно решение

В архива няма официално решение на тази задача.

Оригинал в Архива: exp.pdf