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2 3 Since F cc~. the force will become nine r times smaller. The acceleration is the same as the gravitational field strength. Hence a = GM = GM = J.GM = JI r2 4 ' When they come near the sun they move very fast and so on the average spend most of their time away from the sun and out of sight. 5 6 (2 R)Z 4 R2 4 . The force is mg = 0.40 x 15 = 6.0 N. 7 EP = mV= -2.0 x 1010 x 350 = -7.0 x 1012 J The force does not stay constant as we move from A to B. lt is zero since the gradient at P is zero. > 383 • understand the concept and properties of electric charge • charge a body by electrostatic induction • apply Coulomb's law • undj:!rstand the concept of electric field • work with magnetic fields • determine the direction of the magnetic fields created by straight currents and coils • recognize how magnetic fields exert magnetic forces on moving charges and electric currents ) ) understand the concept of electric potential and electric potential energy understand the connection between electric field lines and equipotential surfaces 18 Electric and magnetic fields GUIDING QUESTIONS • • • What is the electric force between charged particles? What is the magnetic force on moving charged particles? What is the connection between electricity and magnetism? Introduction This chapter examines the properties of electric charge and the phenomena that take place when charge is allowed to move so as to create an electric current. The concept of an electric field is crucial to understanding electric current, as it is the electric field inside a conductor that forces electric charge to move. 18.1 Electric charge, force and field Electric charge is a property of matter. Ordinarily, matter appears electrically neutral, but we can charge a body by friction. For example, take two plastic rods and rub each with a piece of wool. We will find that the two rods now repel each other. If we now rub two glass rods with silk, we find that the glass rods again repel each other, but the charged glass rod attracts the charged plastic rod. We can understand these observations (Figure 18.1) by assuming that: • the process of rubbing involves the transfer of charge from one body to the other • charge can be positive or negative • there is a force between charged bodies that can be attractive or repulsive. Figure 18.1: Two simple experiments to investigate properties of electric charge. Benjamin Franklin (1706-1790) decided to call the sign of the charge on the glass rubbed with silk 'positive'. Much later, when electrons were discovered, it was found that electrons were attracted to the charged glass rod. This means that electrons must have negative charge. But if Franklin had called the charge on the glass rod negative, we would now be calling the electron's charge positive! From experiments with charged objects, we learn that there is a force of attraction between charges of opposite sign and a force of repulsion between charges of the same sign. The magnitude of the force becomes smaller as the distance between the charged bodies increases. Properties of electric charge In ordinary matter, negative charge is a property of particles called electrons. Positive charge is a property of protons, which exist in the nuclei of atoms. (There are many other particles that have charge, but those do not appear in ordinary matter.) The first important property of electric charge is that it is conserved. Like total energy, electric charge cannot be created or destroyed. In any process the total charge cannot change (see worked example 18.1). In solid metals the atoms are fixed in position in a lattice, but there are many 'free' electrons that do not belong to a particular atom. When exposed to an electric field (see later) these electrons can drift in the same direction creating an electric current. In liquids, and especially in gases, positive ions can also transport charge. > 385 ) PHYSICS FOR THE IB DIPLOMA: COURSEBOOK Materials that have many 'free' electrons (Figure 18.2) are called conductors. Materials that very few 'free' electrons, so charge cannot move freely, are called insulators. ....-atom with ~ • ~ 'free' electrons • I • electron cloud \ ~ • • • Figure 18.2: In a conductor there are many 'free' electrons that move around much like molecules of a gas. WORKED EXAMPLE 18.1 Two separated, identical conducting spheres are charged with charges of 4.0 J..LC and -12 J..LC. respectively. The spheres are allowed to touch and then are separated again. Determine the charge on each sphere. Describe the transfer of charge from one sphere to the other. Answer The net charge on the two spheres is 4.0- 12 = -8.0 J..LC. The contact of the two conducting spheres implies that charge will be transferred from one to the other. By symmetry, when the spheres are aUowed to touch, they will end up with the same charge, since they are identical. The total amount of charge on the two spheres after separation must be -8.0 J..lC by charge conservation. When they separate, each will therefore have a charge of -4.0 J..lC. It is negative charge that gets transferred (electrons). The positive charges have fixed positions and do not move. So an amount of -8.0 J..lC gets transferred. The second important property of electric charge is that it is a quantised quantity; this means that the amount of electric charge on a body is always an integral multiple of a basic unit. The basic unit is the magnitude of the charge on the proton. > This amount of charge is symbolised by e. The charge on an electron is - e. (If we include quarks, the particles inside protons and neutrons, then the basic unit of charge is J·) The SI unit of charge is the coulomb (C). I e = 1.6 x IQ-19 C. The quantisation of electric charge was determined in an experiment by Robert Millikan and is described at the end of this chapter. Electrostatic induction Friction is one way to charge a body. Electrostatic induction is another. Consider a positively charged rod that is placed close to, but not touching, a conducting sphere on an insulating stand. .r{) ~ electrons 1--ll-- Figure 18.3: Charging a sphere by induction. The positive charge on the rod will attract free electrons in the sphere closer to the rod. This means the left side of the sphere will be left with an excess of positive charge. The net charge on the sphere is still zero. In the second diagram the sphere is grounded. This means that we connect the sphere to the ground with a cable. Electrons from the earth will move through the cable and neutralize some of the positive charge on the left side of the sphere. The connection to the earth and the charged rod are both removed. The sphere is left with a net negative charge that will distribute itself uniformly on the surface of the sphere. The sphere is left with a net charge that is opposite to that on the charged rod. If the rod is negatively charged, the negative charge will push electrons away leaving the right side of the sphere with an excess of positive charge. Grounding (earthing) will make electrons move to the earth leaving the sphere with a net positive charge after grounding and rod are removed. Again the charge on the sphere is opposite to that on the rod. 386 18 Electric and magnetic fields ' CHECK YOURSELF 1 A student says, referring to Figure 18.3, that when the positively charged rod is put close to the sphere, negative charge is attracted to the right side of the sphere and positive charge is repelled to the left side. Comment on this statement. WORKED EXAMPLE 18.2 Two charges, q1 = 2.0 11C and q2 = 8.0 J.!C, are placed along a straight line separated by a distance of 3.0 cm. a b Coulomb's law for the electric force The electric force between two electric charges, q 1 and q2 , was investigated in 1785 by Charles Augustin Coulomb (1736-1806). Coulomb discovered that this force is inversely proportional to the square of the separation of the charges and is proportional to the product of the two charges. It is attractive for charges of opposite sign and repulsive for charges of the same sign (Figure 18.4). c Calculate the force exerted on each charge m vacuum. Calculate the force when the charges are surrounded by water whose permittivity is 80 times that of the vacuum. The charge q1 is increased to 4.0 11C. Determine the force on each charge now (in vacuum). Answer a b Figure 18.4: The force between two point electric charges is given by Coulomb's law and can be attractive or repulsive. c This is a straightforward application of the formula F = k q,; 2 • We find that: r F = 8.99 X 109 X 2.0 X 8.0 X JO- ll 9.0 X J0-4 F= 160N This is the force that q1 exerts on q2 , and vice versa. The Coulomb constant is now 80 smaller so the force is 2.0 N. Since the charge doubles the force doubles to F = 320 N on both charges. lt is a common mistake to double the force on one charge but not the other. Coulomb's law states that the electric force F between two point charges q 1 and q2 separated by a distance r is given by: F= k q,q2 r l The constant k is known as the Coulomb constant and equals 8.99 x 109 NC-2m2 in vacuum. The constant k is also written ask = -4 1 where Trt: e is known as permittivity. If the charges are surrounded by vacuum, the value of the permittivity is e0 = 8.85 x 10-12 C2N-1 m-2 . If they are surrounded by some other medium, we must use the value of the permittivity appropriate to that medium. WORKED EXAMPLE 18.3 A positive charge q is placed on the line joining q1 and q2 from worked example 2.2. Determine the distance from q1 where this third positive charge experiences zero net force. Answer Let that distance be x. A positive charge qat that point would experience a force from q1 equal to F1 = k q,; and a force in the opposite direction from q2 X qq equal to F2 = k (d ~ x)l where d = 3.0 cm is the distance between q1 and q2 • > 387 ) PHYSICS FOR THE IB DIPLOMA: COURSEBOOK CONTINUED F2 d q F1 q, o----- ~----------------------0 q2 X Figure 18.5: For worked example 18.3. Charge q