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it makes 100 oscillations in 7 min. 30 sec., and when the magnet A is replaced by another B the needle makes 100 oscillations in 6 min. 40 sec. Compare the magnetic moments of A and B.

21. A compass needle makes IO oscillations per minute under the influence of the earth's magnetism alone. When the north pole of a long magnet A is held 1 ft. south of it, the needle makes 12 oscillations per minute; and when the north pole of another magnet B is held in the same position, the number of oscillations per minute increases to 15. Compare the pole-strengths of the magnets A and B.

22. At Berlin the total magnetic intensity is 0.48 (in C.G.S. units) and the dip is 64°: at New York the total intensity is 0.61 and the dip 72°. If a magnet vibrating horizontally at Berlin makes 20 oscillations in a minute, how many oscillations would it make in the same time at New York?

23. At a certain place a dip-needle makes an angle of 60° with the horizontal. When a weight of 1 gm. is attached to the upper end, the inclination is reduced to 30°: what weight would make it horizontal ?

24. A dip-circle is rotated (in azimuth) through an angle a from the magnetic meridian, and the apparent angle of dip under these conditions is ': prove that the true dip (0) at the place is given by the equation

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25. Discuss the precise advantages of the method usually adopted for determining the magnetic dip (i.e. by observing the position in which the needle points vertically downwards, and then rotating the dip-circle. through 90°); and prove that the true dip may be found from observations in any two azimuths at right angles by the formula

cot20=cot2 01 + cot2 2,

01 and 2 being the observed angles of dip in any two planes at right angles, and being the true dip.

EXAMINATION QUESTIONS

26. A bar magnet, hung horizontally by a fine wire, lies in the magnetic meridian when the wire is without twist. It is then found that when the top of the wire is twisted through 120° the magnet is deflected through 30°. Through what further angle must the top of the wire be twisted in order to turn the magnet perpendicular to the magnetic meridian ? S. K. 1893.

To deflect

27. A bar magnet is suspended horizontally in the magnetic meridian by a wire without torsion. the bar 10° from the meridian the top of the wire has to be turned through 180°. The bar is removed, remagnetised, and restored, and the top of the wire has now to be turned through 250° to deflect the bar as much as before. Compare the magnetic moments of the bar before and after remagnetisation.

S. K. 1894.

28. What is meant by the horizontal intensity of the earth's magnetic field?

A horizontally suspended magnet vibrates 12 times per minute at a place where the horizontal intensity is 0.180. How many times per minute will it vibrate at a place where the horizontal intensity is o-245?

Int. Sc. 1893

29. Two bar magnets, the moments of which are in the ratio 8:27, are placed with their centres 3 feet apart, their magnetic axes being in the same straight line, which is perpendicular to the magnetic meridian. If their north poles are turned towards each other, find the position which a small compass needle must occupy on the line joining the magnets in order that it may point in the same direction as if the magnets were not there. S. K. 1889. 30. A magnetic needle makes a complete vibration

in a horizontal plane in 2.5 seconds under the influence of the earth's magnetism only, and when the pole of a long bar-magnet is placed in the magnetic meridian in which the needle lies, and 20 cm. from its centre, a complete vibration is made in 1.5 seconds. Assuming H=.18 (C.G.S.), and neglecting the torsion of the fibre by which the needle is suspended, determine the strength of the pole of the long magnet. Int. Sc. (Hons.) 1886.

31. Two small straight magnets, of magnetic moments M, M', are placed in the same straight line with similar poles facing each other. Show that, if their dimensions are negligible in comparison with r, the distance between their centres, the force tending to separate them is 6MM'r-4.

S. K. 1894.

32. The north pole of a very long vertical magnet whose strength was 250, was placed at a perpendicular distance of 20 cm. from the centre of a horizontal magnetic needle whose length was 5 cm. and strength 50. Find the moment of the couple acting upon the small magnet.

Camb. Schol. 1891.

33. Explain how the law of force between two magnetic poles has been established.

A small galvanometer needle, swinging freely under the earth's magnetic force alone, makes 3 complete oscillations per sec.; the strength of the field in which the needle hangs is diminished by the control magnet of the galvanometer until the time of a complete oscillation is 2 secs. Find how much the "zero-position" of the needle will be altered by a change in the magnetic declination of 1 min. of arc. By how many millimetre-scaledivisions of a reflecting galvanometer will the change of zero be represented, if the scale be 50 cm. from the mirror? N S. Tripos 1890.

R

CHAPTER X

ELECTROSTATICS

Note. All quantities are expressed in terms of the C.G.S. units. For the definitions of the electrostatic units and their dimensions, see pp. 4 and 17.

1. Two small spheres are at a distance of 5 cm. apart one has a charge of 10 units of electricity, the other a charge of 5 units. What is the force exerted between them?

It follows from Coulomb's law, and from the definition of the unit quantity of electricity, that the force (in dynes) is equal to the product of the charges divided by the square of the distance between the spheres.

Thus

F = 10 × 5/52 = 50/25=2 dynes.

If the two charges are of the same kind (i.e. both positive or both negative) the force will be one of repulsion; if the one charge is positive and the other negative, the force will be one of attraction.

2. Two small electrified bodies at a distance of 12 cm. apart are found to attract one another with a force of 6 dynes. The one has a positive charge of 32 units: what is the charge of the other?

3. What is the distance between two small spheres which have charges of 32 and 36 units respectively, and repel one another with a force of 8 dynes?

4. Express in dynes the repulsive force exerted be

tween two small spheres 15 cm. apart, and charged respectively with 40 and 45 units of electricity.

5. Two small spheres are 10 cm. apart, and one of them has a charge of 45 units: what must be the charge on the other so that the force exerted between them may be equal to the weight of 5 milligrammes ?

6. Determine the relation between the electrostatic unit of quantity in the metre-milligramme-minute system and the corresponding C.G.S. unit.

7. An electrified ball is placed in contact with an equal and similar ball which is unelectrified on being separated 8 cm. from one another the force of repulsion between them is equal to 16 dynes. What was the original charge on the electrified ball?

Since the balls are of equal size the charge will be equally shared between them when they are placed in contact. Let q be the charge on each: then the repulsive force between them is (92/82), and this is equal to 16 dynes. Thus q282 x 16, and q=8 × 4 = 32. The original charge on the electrified ball was 2q=2 × 32=64 units.

8. Two small equal balls, one having a positive charge of 10 units and the other a negative charge of 5 units, are 5 cm. apart: what is the attractive force between them? If they are made to touch, and again separated by the same distance, what will be the force of repulsion ?

9. A small uncharged sphere is placed in contact with an equal charged sphere, and then removed to a distance of 6 cm. They repel each other with a force of 4 dynes : what was the original charge? At what distance must they be placed so that the force may be diminished to 1 dyne ?

10. Two small spheres, each charged with 50 units of electricity, are placed at two of the corners of an equilateral triangle 1 metre on the side: what is the magnitude and direction of the resultant electric force at the third corner?

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