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 2MM (1+4), where M and M, are the magnetic 23 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. 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 × 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 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 be the charge on each then the repulsive force between them is (92/82), and this is equal to 16 dynes. Thus q2=82 × 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. 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/π? The surface-density is the quantity of electricity per unit of surface. Thus if S be the area of the surface, and the surface-density, the charge is Q-So. The area of the surface of a sphere of radius is 4πr2: thus S = 4 × (25)2, and Q=4′′ × (25)2 × 5/π = 20 × 625=12500. 11. A sphere of 5 cm. radius has a charge of 1000 units of electricity: what is the surface-density of the charge? 12. What charge must be imparted to a spherical conductor of 3 cm. diameter in order that the superficial density of the electrification may be 7? [Take π= 22/7.] 13. A magnetised knitting-needle, carrying a small gilt pith-ball at one end, is suspended horizontally by a silk fibre a second pith-ball, of the same size as the first, is electrified and brought into contact with it. Prove that the charge on the second pith-ball is proportional to (sin), where a is the angle through which the knitting-needle is deflected from the magnetic meridian. 14. Three small electrified spheres, A, B, and C, have charges 1, 2, and 4 respectively. Find the position in which B must be placed between A and C in order that it may be in equilibrium. Prove also that there is another position along the line CA produced in which B will be equally repelled by A and C. 15. The bob of a seconds pendulum consists of a sphere of mass 16 grammes, and it is suspended by a silk thread. Vertically beneath it is placed a second sphere, which is positively electrified, and when the pendulum-bob is negatively electrified its time of oscillation is found to be 0.8 sec. Prove that the attractive force between the two spheres is equal to the weight of 9 grammes. (The arc of vibration is supposed to be so small that the attractive force is always along the vertical.) Potential and Capacity.-It can be shown 1 that 1 Clerk Maxwell, Elementary Treatise on Electricity, Art. 86; Silvanus Thompson, Electricity and Magnetism, Art. 238. [CH. if a quantity g of electricity be collected at a given point, the difference of potentials due to it at any two given points A and B, whose distances from the given point are and respectively, is VA-VB=q|rq|r'. If the point B is removed to an infinite distance, or is connected with the earth, becomes ∞, and q/r′ = 0. If we agree to regard the potential of the earth as zero, the expression for the difference of potentials between A and the earth, or, more briefly, the potential of the point A, reduces to VA=q/r. If instead of a single quantity of electricity there are several charges 91, 92, 93, . . whose distances from the given point are r 2, 73, ・ ・ ・ respectively, then the potential at A due to all these charges is 9 92193+ VA = + Σ (1) The external action of an electrified spherical conductor is the same as if all the charge were collected at its centre. If the charge be Q, the potential due to it at any external point, whose distance from the centre of the sphere is r, is Q/r. This is only true when ʼn is not less than the radius R of the sphere. At the surface of the sphere r = R, and the potential is Q/R. Now the capacity (C) of the sphere is measured by the charge required to raise its potential from zero to unity, or C=Q/V, where Q is the charge and V the potential due to it. But we have seen that V = Q/R. Hence C=R, or— The capacity of a spherical conductor (placed in air at a considerable distance from other conductors) is numerically equal to its radius. |