1040 Phys Lecture 1. Electric Force. The electromagnetic force between charged particles is one of the fundamental forces of nature.

Transcription

1 Electric Force The electromagnetic force between charged particles is one of the fundamental forces of nature. 1- Properties of Electric Charges A number of simple experiments demonstrate the existence of electric forces and charges. For example, after running a comb through your hair on a dry day, you will find that the comb attracts bits of paper. The attractive force is often strong enough to suspend the paper. The same effect occurs when certain materials are rubbed together, such as glass rubbed with silk or rubber with fur. Figure (1) Another simple experiment is to rub an inflated balloon with wool. The balloon then adheres to a wall, often for hours. When materials behave in this way, they are said to be electrified, or to have become electrically charged. Next Figure shows that "Rubbing a balloon against your hair on a dry day causes the balloon and your hair to become charged ".

2 Figure () In a series of simple experiments, it was found that there are two kinds of electric charges, positive and negative charges. We identify negative charge as that type possessed by electrons and positive charge as that possessed by protons. To verify that there are two types of charge, suppose a hard rubber rod that has been rubbed with fur is suspended by a sewing thread, as shown in Next Figure (3). When a glass rod that has been rubbed with silk is brought near the rubber rod, the two attract each other (Fig. 3-a). On the other hand, if two charged rubber rods (or two charged glass rods) are brought near each other, as shown in Figure 3-b, the two repel each other. This observation shows that the rubber and glass have two different types of charge on them. On the basis of these observations, we conclude that " charges of the same sign repel one another and charges with opposite signs attract one another."

3 Figure (3) Then we can say that "the electric charge on the glass rod is called positive and that on the rubber rod is called negative. Therefore, any charged object attracted to a charged rubber rod (or repelled by a charged glass rod) must have a positive charge, and any charged object repelled by a charged rubber rod (or attracted to a charged glass rod) must have a negative charge". Another important aspect of electricity that arises from experimental observations is that electric charge is always conserved in an isolated system. That is, when one object is rubbed against another, charge is not created in the process. The electrified state is due to a transfer of charge from one object to the other. One object gains some amount of negative charge while the other gains an equal amount of positive charge. For example, when a glass rod is rubbed with silk, as in Figure 1, the silk obtains a negative charge that is equal in magnitude to the positive charge on the glass rod.

4 Also Another important aspect that electric charge always occurs as some integral multiple of a fundamental amount of charge e. In modern terms, the electric charge q is said to be quantized, where q is the standard symbol used for charge as a variable. That is, electric charge exists as discrete packets, and we can write q = Ne, where N is some integer. Other experiments in the same period showed that the electron has a charge -e and the proton has a charge of equal magnitude but opposite sign +e. Some particles, such as the neutron, have no charge. From our discussion thus far, we conclude that electric charge has the following important properties: There are two kinds of charges in nature; charges of opposite sign attract one another and charges of the same sign repel one another. Total charge in an isolated system is conserved. Charge is quantized. - Charging Objects By Induction It is convenient to classify materials in terms of the ability of electrons to move through the material: Electrical conductors are materials in which some of the electrons are free electrons that are not bound to atoms and can move relatively freely through the material. In contrast, materials such as copper, aluminum, and silver are good electrical conductors. When such materials are charged in some small region, the charge readily distributes itself over the entire surface of the material. electrical insulators are materials in which all electrons are bound to atoms and cannot move freely through the material. Materials such as glass, rubber, and wood fall

5 into the category of electrical insulators. When such materials are charged by rubbing, only the area rubbed becomes charged, and the charged particles are unable to move to other regions of the material. Semiconductors are a third class of materials, and their electrical properties are somewhere between those of insulators and those of conductors. Silicon and germanium are well-known examples of semiconductors commonly used in the fabrication of a variety of electronic chips used in computers. 3- Coulomb s Law Charles Coulomb ( ) measured the magnitudes of the electric forces between charged objects and he found that:- The electric force is inversely proportional to the square of the separation r between the 1 particles and directed along the line joining them F ; r The electric force is proportional to the product of the charges q1 and q on the two particles F q q ; 1 The electric force is attractive if the charges are of opposite sign and repulsive if the charges have the same sign; The electric force is a conservative force. We will use the term point charge to mean a particle of zero size that carries an electric charge. We can express Coulomb s law as an equation giving the magnitude of the electric force (sometimes called the Coulomb force) between two point charges: F e = K Where k e is a constant called the Coulomb constant. The value of the Coulomb constant depends on the choice of units. The Coulomb constant ke in SI units has the value e q q 1 r

6 K 1 = 4 πε 0 = N.m /C where the constant ε o (lowercase Greek epsilon) is known as the permittivity of free space and has the value ε o = The smallest unit of charge e known in nature is the charge on an electron (- e) or a proton (+ e) and has a magnitude e = 1.60 x 10 C 19 / N.m C Example 1 (The Hydrogen Atom) The electron and proton of a hydrogen atom are separated (on the average) by a distance of approximately 5.3 x m. Find the magnitudes of the electric force and the gravitational force between the two particles. Solution From Coulomb s law, we find that the magnitude of the electric force is - m + النواة M

7 F e = K = = e q q r 9 x = K x 8. x 10 8 e 19 ( 1.6 x 10 ) 11 ( 5.3 x 10 ) Using Newton s law of universal gravitation and above Table for the particle masses, we find that the magnitude of the gravitational force is = N e mm F G = G r x r e ( N.m /Kg )( Kg) ( Kg) 11 ( m) = N F F E G = = Thus, the gravitational force between charged atomic particles is negligible when compared with the electric force. Figure (4)

8 When dealing with Coulomb s law, you must remember that force is a vector quantity and must be treated accordingly. The law expressed in vector form for the electric force exerted by a charge q 1 on a second charge q, written F 1, is q 1q = K r r F1 e where rˆ is a unit vector directed from q 1 toward q, as shown in Figure 4-a. Because the electric force obeys Newton s third law, the electric force exerted by q on q 1 is equal in magnitude to the force exerted by q 1 on q and in the opposite direction; that is, F 1 = - F 1. Figure (5) if five charges are present, as shown in figure(5), then the resultant force exerted by particles, 3, 4,and 5 on particle 1 is F + 1 = F1 + F31 + F41 F51

9 Example Find the Resultant Force Consider three point charges located at the corners of a right triangle as shown in Figure 6, where q 1 = q 3 = 5.0 µc, q = -.0 µc, and a = 0.10 m. Find the resultant force exerted on q 3. y F 13 F 13 y + q 3 F 13 x x Figure (6) Solution First, note the direction of the individual forces exerted by q 1 and q on q 3. The force F 3 exerted by q on q 3 is attractive because q and q 3 have opposite signs. The force F 13 exerted by q 1 on q 3 is repulsive because both charges are positive. The magnitude of F 3 is

10 In the coordinate system shown in Figure(6), the attractive force F 3 is to the left (in the negative x direction). The magnitude of the force F 13 exerted by q 1 on q 3 is The repulsive force F 13 makes an angle of 45 with the x axis. Therefore, the x and y components of F 13 are equal, with magnitude given by F 13 cos 45 = 7.9 N. We can also express the resultant force acting on q 3 in unit vector form as Figure (7)

11 Example 3 Where Is the Resultant Force Zero? Three point charges lie along the x axis as shown in Figure (7). The positive charge q 1 = 15.0 µc is at x =.00 m, the positive charge q = 6.00 µc is at the origin, and the resultant force acting on q 3 is zero. What is the x coordinate of q 3? Solution Because q 3 is negative and q 1 and q are positive, the forces F 13 and F 3 are both attractive, as indicated in Figure (7). From Coulomb s law, F 13 and F 3 have magnitudes For the resultant force on q 3 to be zero, F 3 must be equal in magnitude and opposite in direction to F 13. Setting the magnitudes of the two forces equal, we have Noting that k e and q 3 are common to both sides and so can be dropped, we solve for x and find that Solving this quadratic equation for x, we find that the positive root is x = m There is also a second root, x = m. This is another location at which the magnitudes of the forces on q 3 are equal, but both forces are in the same direction at this location.

12 Example 4 Find the Charge on the Spheres Two identical small charged spheres, each having a mass of 3.0 x 10 - kg, hang in equilibrium as shown in Figure (8) -a. The length of each string is 0.15 m, and the angle θ is 5.0. Find the magnitude of the charge on each sphere. Figure (8) The sphere is in equilibrium under the application of the forces T from the string, the electric force F e from the other sphere, and the gravitational force mg. Because the sphere is in equilibrium, the forces in the horizontal and vertical directions must separately add up to zero:

13 Considering the geometry of the right triangle in Figure 8-a, we see that sin θ = a/l. Therefore, The separation of the spheres is a = 0.06 m. From Coulomb s law, the magnitude of the electric force is where r = a = 0.06 m and q is the magnitude of the charge on each sphere. References This lecture is a part of chapter 3 from the following book Physics for Scientists and Engineers (with Physics NOW and InfoTrac), Raymond A. Serway - Emeritus, James Madison University, Thomson Brooks/Cole 004, 6th Edition, 196 pages.

14 Problems (1) Two protons in an atomic nucleus are typically separated by a distance of x m. The electric repulsion force between the protons is huge, but the attractive nuclear force is even stronger and keeps the nucleus from bursting apart. What is the magnitude of the electric force between two protons separated by.00 x m? () Two protons in a molecule are separated by 3.80 x m. Find the electric force exerted by one proton on the other. (b) How does the magnitude of this force compare to the magnitude of the gravitational force between the two protons? (c) What If? What must be the charge to mass ratio of a particle if the magnitude of the gravitational force between two of these particles equals the magnitude of electric force between them? (3) Three point charges are located at the corners of an equilateral triangle as shown in Figure 9. Calculate the resultant electric force on the µc charge. Figure (9) (4) Two identical conducting small spheres are placed with their centers 0.300m apart. One is given a charge of 1.0 nc and the other a charge of nc. (a) Find the electric force exerted by one sphere on the other. (b) What If? The spheres are connected by a conducting wire. Find the electric force between the two after they have come to equilibrium.

15 Problems solutions (1) ()

16 (3) (4)