ELECTRIC CHARGE Introuction: Orinary matter consists of atoms. Each atom consists of a nucleus, consisting of protons an neutrons, surroune by a number of electrons. In electricity, the electric charge (q, Q) plays the same rule as mass oes in mechanics. The unit of electric charge q is the Coulomb (C). In nature there are two kins of charges, positive an negative, with the properties that: 1. The force between the charge particles is calle the electric force.. Unlike charges, opposite electrical sign, attract one another, an like charges, same electrical sign, repel one another, see the following figures. 3. The force between stationary charges is calle electrostatic force. The electrostatic force obeys Coulomb s law. 4. The SI unit of charge is the Coulomb, C. 5. Electric charge is quantize, by mean that if the magnitue of the smallest charge is enote by e (calle the charge quantum), where e = 1.6 10-19 C, then all other charges are integer multiplies of e, i.e. q n e, n 1,,3, Although there is a goo reason to believe that charge magnitues of e/3 an e/3 are possible, extensive experiments have not yet verifie their existence. 6. An uncharge object becomes negatively charge if it gains electrons an becomes positively charge if it loses electrons. All electrons carry the same quantity of charge, so the more electrons an object gains or loses the greater is the overall negative or positive charge upon it. 7. Also, the algebraic sum of the charges in the universe is constant (conservation of charge). When a particle with charge +e is create, a particle with charge -e is create simultaneously. When a particle with charge +e isappears, a particle with -e also isappears. Hence the net charge of the universe remains constant. The masses an charges of the electrons, protons an neutrons are liste in following table. Most of the mass of the atom is ue to the mass of the nucleus. particle mass (kg) charge (C) electron 9.11 x 10-31 - 1.6 x 10-19 proton 1.673 x 10-7 1.6 x 10-19 neutron 1.675 x 10-7 0 Masses an charges of the builing blocks of atoms. The iameter of the nucleus is between 10-15 an 10-14 m. The electrons are containe in a roughly spherical region with a iameter of about x 10-10 m. Measurements of the velocity of the orbital electrons in an atom have shown that the attractive force between the electrons an the nucleus is significantly stronger than the gravitational force between these two objects. 1
Coulomb s law: "The magnitue of the electric force, F, that a charge particle, q, exerts on another particle, Q, is irectly proportional to the prouct of their charges an inversely proportional to the square of the istance between them. The irection of the force is along the line joining the particles." Mathematically: q Q 1 N.m² 9 F k, k 9 10 r ² 4, o C where o is the permittivity of free space, o 9 10-1 C²/(N.m²). Coulomb's law applies to elementary particles an small charge objects as long as their sizes are much less than the istance between them. It is also applies to uniform spherical shells or spheres of charge. In that case, r, the istance between the centers of the spheres, must be larger than the sum of the sphere raii; that is to say, the charge must be largely separate. The electrostatic force between two point charges can be expresse in vector form. The force on ue to is written as: q 1 where r 1 q F 1 qq k r 1 1 is a unit vector irecte from rˆ q 1 1 to q an r 1 is the istance between them.. When more than one point charge is present, the total electrostatic force exerte on the i th charge is the vector sum of the forces exerte on that charge by others iniviually, i.e. q j F i = Fij kqi r j r where r j j = 1 i < j is a vector irecte from the charge, q i i j q1 ˆr 1, to the point in question. r1 q Two ientical positive charges Q are on the F1 Q -a a Q F x-axis at a an +a. Fin the magnitue an irection of the net force on each charge. Answer: (i) The net force acting on the charge at a is: QQ Q F1 k k, to the negative x-axis. ( a) 4a The net force acting on the charge at a is: QQ Q F k k, to the positive x-axis. ( a) q1 4a x 1 m q3 q
Two charges one of of a thir charge of If the thir charge, q1 C q3 3 C q 3 an one of q 8 C which feels no net force on it. are one meter apart. Fin the location, is locate between the two charges, the effective repulsive forces it feels will cancel each other. Suppose x is the istance of F F 13 3 q 3 from q 1, then: q q q q q x k 1 3 3 (1 ) 8 k x (1 x ) q1 x Taking the square root, one can have: (1 x ) x x 1/ 3, 1 So, the answer is 0.33 m from q 1. accepte Notice that the point of equilibrium oes not epen on the charge H. W. Two charges one of location (from q1 9 C q 1 ) of a thir charge q 3 rejecte q 3, but on the other charges. an one of q 4 C are two meters apart. Fin the which feels no net force on it. [Ans: 6 m] Three point charges of µc, 7 µc, an - 4 µc are locate at the corners of an equilateral triangle of sie length 0.5 m. Calculate the net force on the 7 µc charge. 6 6 9 (10 )(710 ) 7, F (910 ) 0.504 N (repulsive) (0.5) 6 6 9 (410 )(710 ) 7, 4 F (910 ) 1.008 N (attractive) (0.5) o F F F cos 60 0.756 N, x 7, 7, 4 7, 7, 4 o F F F sin 60 0.436 N, y As a vector F (0766 i 0.436 j) N o F = 0.873 N, at an angle of 330 with the positive x-axis. y F7, 7 µc x F7,-4 µc -4 µc Example: What is the net force experience by the charge at point P? 3
Solution: The force experience by the +Q charge from the other three charges are inicate below: Note that the magnitues of the contributions from 1 an are the same since charges 1 an are the same. A irection has been assigne to the contribution from charge 3 [iagonally inwar] but its magnitue relative to the others is not known until we evaluate it. Recognizing that the resultant of the contributions from 1 an points iagonally outwar, as shown below. The solution to this question reuces to a etermination of the relative magnitue of the contribution from charge 3 to the resultant of the contributions from 1 an. The relationship for the force between two charges: F qq k r i j ij Thus the magnitue of the contribution from 1 (an ) is an the resultant is F F 1 F F 1 The magnitue of the contribution from 3 is F k k Q Q a k Q a a Q Q Q k a 3 which is one-half that from 1 an combine, so the relative contributions shown in the previous figure are correct an the irection of F from all three contributions points iagonally outwar (5 egrees or southwest). 4
Conuctors an Insulators A conuctor is a material that permits the motion of electric charge through its volume. Examples of conuctors are copper, aluminum an iron. An electric charge place on the en of a conuctor will sprea out over the entire conuctor until an equilibrium istribution is establishe. In contrast, electric charge place on an insulator stays in place: an insulator (like glass, rubber an Mylar) oes not permit the motion of electric charge. The properties of a conuctor are a result of the presence of free electrons in the material. These electrons are free to move through the entire volume of the conuctor. Because of the free electrons, the charge istribution of a conuctor can be change by the presence of external charges. For example, the metal sphere shown in the figure is initially uncharge. This implies that the free electrons (an positive ions) are istribute uniformly over its surface. If a ro with a positive charge is place in the vicinity of the sphere, it will prouce an attractive force on the free electrons. As a consequence of this attractive force the free electrons will be reistribute, an the top of the conuctor will get a negative charge (excess of electrons). Since the number of free electrons on the sphere is unchange, the bottom of the sphere will have a eficit of free electrons (an will have a positive charge). The positive ions are boun to the lattice of the material, an their istribution is not affecte by the presence of the charge ro. If we connect the bottom of the sphere to groun (a source or rain of electrons) electrons will be attracte by the positive charge. The number of electrons on the sphere will increase, an the sphere will have a net negative charge. If we break the connection to the groun before removing the charge ro, we are left with a negative charge on the sphere. If we first remove the charge ro, the excess of electrons will rain to the groun, an the sphere will become uncharge. 5
Suppose that isolate charges Q an q attract each other with a force F. If the separation between these charges were mae half as great, each charge woul then experience a force (a) F/3. (b) F. (c) Can not be etermine unless we know the magnitue of Q an q. ()@ 4 F. (e) F/. Three charges are locate as shown in Figure 1. If a = 3.0 m, Q1 =.0 C, an Q = Q3 = 8.0 C, what is the magnitue of the electric force on charge Q1? (a) 0.03 N (b) 0.090 N (c) 0.046 N ()@ 0.011 N (e) 0.055 N Three points charges are locate on the x-y plane as follows: Q1 = -10 = 0 x-irection, fin the charge Q3. (a) -4 (b) +16 C (c) 0 ()@ -16 C (e) +4 C C at (4 m, 0), Q C at (0, 10 m), an Q3 at (4 m, 10 m). If the net force on Q1 points in the negative C C A 0. g metallic ball hangs from an insulating string in a vertical electric fiel of 3000 N/C an irecte upwar as shown in Figure 1. If the tension in the string is 0.004 N, then the charge on the ball is: 6
(a) -1.0 (b)@ -0.7 (c) 1.0 C () -.0 (e) 0.7 C C C C A charge +q is place at the origin an a charge -q is place at x= 0.00 m on the x-axis. Where, on the x-axis, can a thir charge +q be place so that the force on it is zero? (a) 0.37 m (b) -0.740 m (c) -0.440 m () 0.11 m (e)@ 0.683 m Consier three point charges, Q1=Q= 1. Fin the magnitue of the resultant force on Q3. C an Q3= 4 C, locate as shown in Figure (a) 10*10**(-3) N (b).9*10**(-3) N (c) 6.0*10**(-3) N ()@ 8.5*10**(-3) N (e) zero A negative charge is place at the center of a square. Each corner of the square has a fixe charge of 1.00 C. If the resulting force acting on each charge is zero, the magnitue of the negative charge is: (a) 9.60 C. (b) 0.77 C. (c) 0.69 C. 7
()@ 0.96 C. (e) 6.9 C. Two neutral metal sphere are separate by 0.3 km. How much electric charge must be transferre from one sphere to the other so that their electrical attraction is 10**3 N? (a) 0. C. (b) 0.9 C. (c) 0.4 C. () 0.6 C. (e)@ 0.1 C. A charge of + 3. If a charge of 1.0 (a) + 0. (b) - 3.3 (c)@ - 0. () +.1 (e) -.1 C C C C C C is place at the origin. A secon charge (q) is place at x = 3.0 m. C experiences no force if place at x = 4.0 m, then q is: Two small charge objects repel each other with a force F when separate by a istance. If the charge on each object is reuce to one-fourth of its original value an the istance between them is reuce to / the force becomes: (a) F. (b)@ F/4. (c) F/16. () F/8. (e) F/. Two fixe particles, of charges q1 = + 1.0 C, are 10 cm apart. How far from each shoul a thir charge be locate so that no net electrostatic force acts on it? (a) 1.1 cm from q1 an 11.1 cm form q. (b)@ 5.0 cm from q1 an 15.0 cm from q. (c) 3.0 cm from q1 an 7.0 cm from q. () 1.0 cm from q1 an 11.0 cm from q. (e) 1.0 cm from q1 an 9.0 cm from q. C an q= - 9.0 A mass with a charge "Q" is suspene in equilibrium from a beam balance. A point charge q = + 10 C is then fixe at a istance = 5.0 cm below "Q" an an extra mass m = 4.0 g has to be place on the pan to obtain equilibrium, see figure (3). Fin the value of the charge "Q". 8
(a) + 6.*10**(-9) C. (b)@ - 1.1*10**(-9) C. (c) + 3.3*10**(-9) C. () + 1.1*10**(-9) C. (e) - 3.3*10**(-9) C. Charges q1 an q are on the x-axis. q1 is at x = a an q is at x = a. The net force on a thir charge at the origin is zero. Which of the following is TRUE? (a)@ q = -4*q1 (b) q = *q1 (c) q = 4*q1 () q = -q1 (e) q = -*q1 Two point charges q1 an q lie along the x-axis. q1 = + 16.0 C is at x =.00 m an q = + 9.00 such that the net electrostatic force on it is zero? (a)@ x = + 0.857 m (b) x = - 0.857 m (c) x = + 1.14 m () x = +.86 m (e) x = - 1.14 m C is at the origin. Where must a negative charge q3 be place on the x-axis As in figure (1), a charge Q is fixe at each of two opposite corners of a square. A charge q is fixe at each of the other two corners. If the resultant electrical force on Q is zero, then Q an q are relate as: (a) Q = - Sqrt() q** (b)@ Q = - Sqrt() q (c) Q = q** () Q = q 9
(e) Q = - 4 q Consier two ientical conuctor spheres, A an B. Initially, sphere A has a charge of -80 Q an Sphere B has a charge of +0 Q. If the spheres touche an then are separate by a istance of 0.3 m, what is the resultant force between them? [Take Q = 5.7*10**(-8) C] (a)@ 0.3 N, repulsive. (b) 0.4 N, attractive. (c) 0. N, attractive. () 0. N, repulsive. (e) 0.3 N, attractive. In figure 3, Q = 60 micro-c, q = 0 micro-c, a = 3.0 m, an b = 4.0 m. Calculate the total electric force on q. [i an j are the unit vectors in the positive irection of x-axis an y- axis, respectively]. (a) 1.1 j (N). (b) -0.34 i (N). (c) -0.69 i (N). () 0.34 i (N). (e)@ 0.69 i (N). In figure (1), if Q = 30 C, q = 5.0 the unit vectors in the positive irection of x-axis an y-axis, respectively]. C an = 0.3 m, fin the net force on q. [i an j are (a) -7.5 i (N). (b) 7.5 i (N). (c) 3.8 i (N). () -3.8 j (N). (e)@ zero. What is the electric force between two protons which are separate by 1.6*10**(-15) m. (a) (b) (c) () zero. 90 N, attractive.. N, repulsive.. N, attractive. 10
(e)@ 90 N, repulsive. Two positive charges (+8.0 C an +.0 C) are separate by 300 m. A thir charge is place a istance r from the +8.0 C charge so that the resultant electric force on the thir charge ue to the other two charges is zero. The istance r is: (a) 500 m. (b) 100 m. (c) 300 m. ()@ 00 m. (e) 400 m. Two positively charge particles q1 an q (with q>q1) are fixe in place on the x-axis at the positions shown in figure 1. A thir charge q3 is to be place somewhere on the x- axis such that the net electrostatic force on q3 is zero. Which one of the following statements is TRUE? (a) q3 shoul be place at a point between q1 an q but closer to q. (b) q3 shoul be place to the left of q1. (c) q3 shoul be place to the right of q. () q3 shoul be place at the mi point between q1 an q. (e)@ q3 shoul be place at a point between q1 an q but closer to q1 Consier the arrangement in Fig. 1. If the charge ro is approaching the neutral metal ro without touching it, which of the following is correct? (a)@ (b) (c) () (e) The ball will be repelle to the right The ball will be attracte to the left The ball will stay hanging vertically The neutral metal ro will be pushe up The neutral metal ro will be pushe own. 11
Supplementary Problems q q Q q Q q q q Q FIGURE 1 FIGURE 1- A negative charge Q is place at the center of a square, see figure (1). Each corner of the square has a fixe charge of q = 1.00 x10-6 C. If the resulting force acting on each charge is zero, the magnitue of the negative charge is: Ans: 0.96x10-6 C. - Two neutral metal sphere are separate by 0.3 km. How much electric charge must be transferre from one sphere to the other so that their electrical attraction is 10 3 N? Ans: 0.1 C. 3- As in figure (), a charge Q is fixe at each of two opposite corners of a square. A charge q is fixe at each of the other two corners. If the resultant electrical force on Q is zero, then Q an q are relate as: Ans: Q = - Sqrt() q 4- Consier two ientical conuctor spheres, A an B. Initially, sphere A has a charge of -80 Q an Sphere B has a charge of +0 Q. If the spheres touche an then are separate by a istance of 0.3 m, what is the resultant force between them? [Take Q = 5.7 x10-8 C] Ans:0.3 N. 5- A.0 micro-c charge is place at the origin. An ientical charge is place.0 m from the origin on the x-axis, an a thir ientical charge is place m from the origin on the y-axis. The magnitue of the force on the charge at the origin is: [1.3*10**(-) N] 1
Question 1 Two point charges q1 an q lie along the x-axis. q1 = + 16.0 micro-coulombs is at x =.00 m an q = + 9.00 micro-coulombs is at the origin. Where must a negative charge q3 be place on the x-axis such that the net electrostatic force on it is zero? a x = + 1.14 m b x = - 1.14 m c x = + 0.857 m x = - 0.857 m e x = +.86 m ============================================== Question As in figure (1), a charge Q is fixe at each of two opposite corners of a square. A charge q is fixe at each of the other two corners. If the resultant electrical force on Q is zero, then Q an q are relate as:` a Q = - Sqrt() q** b Q = q c Q = - Sqrt() q Q = q** e Q = - 4 q ============================================== Question 3 Two fixe particles, of charges q1 = + 1.0*10**(-6) C an q= - 9.0*10**(-6) C, are 10 cm apart. How far from each shoul a thir charge be locate so that no net electrostatic force acts on it? 13
a 1 cm from q1 an 11 cm from q. b 5 cm from q1 an 15 cm from q. c 3 cm from q1 an 7 cm from q. 1 cm from q1 an 9 cm from q. e 1.1 cm from q1 an 11.1 cm form q. ============================================== Question 4 Consier two ientical conuctor spheres, A an B. Initially, sphere A has a charge of -80 Q an Sphere B has a charge of +0 Q. If the spheres touche an then are separate by a istance of 0.3 m, what is the resultant force between them? [Take Q = 5.7*10**(-8) C] a 0. N. b 0.3 N. c 0.4 N. 0.6 N. e 0.9 N. ============================================== Question 5 Charges q1 an q are on the x-axis. q1 is at x = a an q is at x = a. The net force on a thir charge at the origin is zero. Which of the following is TRUE? a q = -4*q1 b q = *q1 c q = -q1 q = 4*q1 e q = -*q1 ============================================== Answers 1 c c 3 b 4 b 5 a 14