Problems set # 2 Physics 169 February 11, 2015

Transcription

1 Prof. Anchordoqui Problems set # 2 Phsics 169 Februar 11, Figure 1 shows the electric field lines for two point charges separated b a small distance. (i) Determine the ratio q 1 /q 2. (ii) What are the signs of q 1 and q 2? 2. An ion milling machine uses a beam of gallium ions (m = 70u) to carve microstructures from a target. A region of uniform electric field between parallel sheets of charge is used for precise control of the beam direction. Single ionized gallium atoms with initiall horizontal velocit of m/s enter a 2.0 cm-long region of uniform electric field which points verticall upward, as shown in Fig. 2. The ions are redirected b the field, and eit the region at the angle θ shown. If the field is set to a value of E = 90 N/C, what is the eit angle θ? 3. Two 2.0-g spheres are suspended b 10.0-cm-long light strings, see Fig. 3. A uniform electric field is applied in the direction. If the spheres have charges of C and C, determine the electric field intensit that enables the spheres to be in equilibrium at θ = Three charges of equal magnitude q are fied in position at the vertices of an equilateral triangle (Fig. 4). A fourth charge Q is free to move along the positive ais under the influence of the forces eerted b the three fied charges. Find a value for s for which Q is in equilibrium. You will need to solve a transcendental equation. 5. Eight solid plastic cubes, each 3.00 cm on each edge, are glued together to form each one of the objects (i, ii, iii, iv) shown in Fig. 5. (a) Assuming each object carries charge with uniform densit 400 nc/m 3 throughout its volume, find the charge of each object. (b) Assuming each object carries charge with uniform densit 15.0 nc/m 2 everwhere on its eposed surface, find the charge on each object. (c) Assuming charge is placed onl on the edges where perpendicular surfaces meet, with uniform densit 80.0 pc/m, find the charge of each object. 6. (i) Consider a uniforml charged thin-walled right circular clindrical shell having total charge Q, radius R, and height h. Determine the electric field at a point a distance d from the right side of the clinder as shown in Fig. 6. [Hint: Use the result of Eample 2 given in lecture 2 and treat the clinder as a collection of ring charges.] (ii) Consider now a solid clinder with the same dimensions and carring the same charge, uniforml distributed through its volume. Use the result of Eample 3 given in lecture 2 to find the field it creates at the same point. 7. A uniforml charged rod of length 14.0 cm is bent into the shape of a semicircle as shown in Fig. 7. The rod has a total charge of 7.50 µc. Find the magnitude and direction of the electric field at O, the center of the semicircle. 8. A line of charge with uniform densit 35.0 nc/m lies along the line = 15.0 cm, between the points with coordinates = 0 and = 40.0 cm. Find the electric field it creates at the origin. 9. (i) A rod of length l has a uniform positive charge per unit length λ and a total charge Q. Calculate the electric field at a point P that is located along the long ais of the rod and a distance a from one end. (ii) Identical thin rods of length 2a carr equal charges +Q uniforml distributed along their lengths. The rods lie along the ais with their centers separated b a distance b > 2a (Fig. 8). Show that the magnitude of the force eerted b the left rod on the right one is given b F = 1 4πɛ 0 Q 2 4a 2 ln ( b 2 b 2 4a 2 ).

2 q 2 q 1 Figure 1: Problem 1. P An electric dipole in a uniform electric field is displaced slightl from its equilibrium position, as shown in Fig. 9, where θis small. The separation of the charges is 2a, and the moment of inertia of the dipole is I. Assuming the dipole is released from this position, show that its angular orientation ehibits simple harmonic motion with a frequenc f = 1 2π 2qaE I.

3 Two small spheres, each of mass 2.00 v g, are suspended b v 0 E θ light strings 10.0 cm in length (Fig. P23.62). A uniform +e electric field is applied in the direction. The spheres have charges equal to C and 5.00 Figure 2: Problem C. Determine the electric field that enables the singl-ionized gallium atom has a charge of q =+e, and the mass of m = 70 u means 70 atomic mass its, spheres where one atomic to be massin unitequilibrium is 1 u= at 27 kg. an angle hat we reall have here is a particle under the influence of a constant force, just as if we were to throw ball horizontall and watch its trajector under the influence of gravit (the onl di erence is that nce we have negative charges, things can fall up"). To start with, we will place the origin at the ion s itial position, let the positive ai run to the right, and let the positive ais run straight up. Thus, e particle starts with a velocit purel in the direction: ~v 0 =v ˆ. hile the particle is in the electric-field-containing region, it will eperience a force pointing along the direction, with a constant magnitude of qe. Since the force acts onl in the direction, there will a net acceleration onl in the direction, and the velocit in the direction will remain constant. nce outside the region, the particle will eperience no net force, and it will therefore continue along in straight line. It will have acquired a component θ to θ its velocit due to the electric force, but the mponent will still be v. Thus, the particle eits the region with velocit ~v = v ˆ + v ŷ. The angle which the particle eits the plates, measured with respect to the ais, must be tan = v v hus, just like in an mechanics problem, finding the angle is reduced to a problem of finding the final locit components, of which we alread know one. So, how do we find the final velocit in the rection? Initiall, there is no velocit in the direction, and while E the particle is traveling between e plates, there is a net force of qe in the direction. Thus, the particle eperiences an acceleration a = F m = qe m + Figure 3: Problem P23.62 he electric field is purel in the direction in this case, so E =90 N/C. Now we know the acceleration the direction, so if we can find out the time the particle takes to transit the plates, we are done, nce the the transit time Dt and acceleration a determine v :. A line of positive charge is formed into a semicircle of radius R 60.0 cm as shown in Figure P The charge

4 the magnitude b the charge Problem 63 in al charges Q. The rods lie d b a distance de of the force ven b Figure P Three charges of equal magnitude q are fied in position at the vertices of an equilateral triangle (Fig. P23.64). A fourth charge Q is free to move along the positive ais +q Problems 733 b + a hin-walled right circuarge Q, radius R, and d at a point a distance er as shown in Figure of Eample 23.8 and ing charges.) (b) What with the same dimen-, uniforml distributed f Eample 23.9 to find 36. Three solid plastic clinders a all a 2 have rradius 2.50 cm and length 6.00 cm. One (a) carries charge with uniform densit 15.0 nc/m 2 everwhere on its θ surface. Another (b) carries charge with the same uniform densit on its curved q lateral surface onl. The third (c) carries +Q charge with uniform densit a 3500 nc/m 3 throughout s the plastic. Find the charge of each 2 clinder. a 37. Eight solid plastic cubes, each cm on each edge, are glued together to form each one of the objects (i, ii, iii, and iv) shown in Figure P (a) Assuming each object carries charge with uniform densit 400+qnC/m 3 throughout its volume, find the charge of each object. (b) Assuming each object carries charge Figure with P23.64 uniform densit 15.0 nc/m 2 everwhere on its Figure eposed 4: Problem surface, find 4. the charge on each object. (c) Assuming charge is placed onl on the edges where perpendicular surfaces meet, with uniform densit 80.0 pc/m, find the charge of each object. d (i) (ii) (iii) (iv) Figure P23.37 Figure 5: Problem 5. charge per unit length igure P (a) Show e from the rod along component and is t If? Using our result Section 23.6 Electric Field Lines 38. A positivel charged disk has a uniform charge per unit area as described in Eample Sketch the electric field lines in a plane perpendicular to the plane of the disk passing through its center.

5 sions and carring the same charge, uniforml distributed through its volume. Use the result of Eample 23.9 to find the field it creates at the same point. h R d d Figure P23.34 Figure 6: Problem 6. ( 35. A thin rod of length and uniform charge per unit length lies along the ais, as shown in Figure P (a) Show that the electric field at P, a distance from the rod along its perpendicular bisector, has no component and is given b E 2k e sin 0 /. (b) What If? Using our result to part (a), show that the field of a rod of infinite length is E 2k e /. (Suggestion: First calculate the field at P due to an element of length d, which has a charge d. Then change variables from to, using the relationships tan and d sec 2 d, and integrate over.) O Sectio 38. A p are lin pas 39. A wit fiel 40. Fig cha rat P θ 0 θ Figure 7: Problem 7. P23.33

6 uniforml distributed along their lengths. The rods lie along the ais with their centers separated b a distance b 2a (Fig. P23.61). Show that the magnitude of the force eerted b the left rod on the right one is given b F k e Q 2 4a 2 ln b 2 2 b 2 4a A fourth char q 738 C HAPTER 23 Electric Fields b a a b a b + a position, show that its Figure angular P23.61 orientation ehibits simple Figure 8: Problem 9. harmonic motion with a frequenc f 1 2 2qa E I q 2a + θ q E Figure P23.73 Figure 9: Problem 10. Answers to Quick Quizzes 23.1 (b). The amount of charge present in the isolated sstem after rubbing is the same as that before because charge is