vibrating horizontally at Berlin makes 20 oscillations in a minute, how many oscillations would it make in the same time at New York ? 19. 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 O' : prove that the true dip () at the place is given by the equation tan 0 =tan o' cos a. 20. 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 cot2 0 = cota 04 + cota 02, 0, and 0, being the observed angles of dip in any two planes at right angles, and 0 being the true dip. EXAMINATION QUESTIONS. 21. Explain what is meant by the strength of a magnetic pole, and describe experiments to determine the law of force between two poles. A circle is described round a small magnet in a plane which contains the axis of the magnet, with its centre coincident with that of the magnet. Discuss the changes in the intensity and direction of the force on a magnetic pole which is carried round the circle ; and show how from observations at the points in which the axis of the magnet cuts the circle, and at the points on the circle midway between these, the law of force may be determined. Camb. Schol. 1885. 22. The centre of gravity of a dip needle does not quite coincide with its axis of suspension. Describe the operations necessary in order to eliminate the error which would otherwise arise in the measurement of the magnetic dip. Int. Sc. 1883 23. The intensities of the earth's horizontal magnetic force at two different places can be compared by observing at each the deflection by the same magnet of a small compass-needle placed in the same position relatively to the magnet. Explain the method, and show how the result of the comparison would be effected by a diminution of the magnetic moment of the compass-needle or of the magnet respectively, occurring between the observations at the first station and those at the second. Int. Sc. 1885. 24. Describe the principle of measurement employed in the Torsion-balance. A magnet suspended by a fine vertical wire hangs in the magnetic meridian when the wire is untwisted. If on turning the upper end of the wire half round the magnet is deflected through 30° from the meridian, show how much the upper end of the wire must be turned in order to deflect the magnet 45° and 60° respectively. Int. Sc. 1884. 25. A bar-magnet is suspended in a Torsion-balance by a wire without torsion. When the torsion head is turned through 360°, the bar is deflected 30° from the meridian. Through how many degrees must the torsion head be turned that the magnet may be in equilibrium at right angles to the meridian ? Prel. Sc. 1887. 26. 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. Honours 1886. 27. Two small magnetic needles are placed with their + centres at a distance r (great compared with their own lengths) from each other, so that the centre of one is in the prolongation of the axis of the other. Show that the couple exerted by one on the other is approximately equal to 20,M? (3+4), where M and M, are the magnetic moments of the needles, and A is a constant. B. Sc. 1879 28. Describe the magnetic behaviour of a piece of soft iron in a magnetic field of gradually increasing intensity, and give experimental methods by which the truth of your statements can be verified. B. Sc. Honours 1884. CHAPTER IX 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 16. 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 x 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 between two small spheres 15 cm. apart, and charged respectively with 40 and 45 units 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 chårge 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 (24/82), and this is equal to 16 dynes. Thus q2 = 82 x 16, and q=8 x 4= 32. The original charge on the electrified ball was 29=2 x 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. Two small spheres, each charged with 50 units of electricity, are placed at two of the corners of an equilateral triangle i metre on the side: what is the magnitude and direction of the resultant electric force at the third corner ? 10. What charge is required to electrify a sphere of 25 cm. radius until the surface-density of the electrification is 5/? The surface-density is the quantity of electricity per unit of surface. Thus if S be the area of the surface, and o the surface-density, the charge is Q=So. The area of the |