Homework #4 203-1-1721 Physics 2 for Students of Mechanical Engineering Part A 2. Derive an expression for the work required by an external agent to put the four charges together as indicated in Fig. 28-28 below. Each side of the square as length a. 6. Two parallel, flat, conducting surfaces of spacing d = 1.0 cm have a potential difference V of 10.3 kv. An electron is projected (launched) from one plate directly toward the second. What is the initial velocity (v i ) of the electron if it comes to rest just at the surface of the second plate? 10. An electron is projected with an initial speed of v i = 3.44 x 10 5 m/s directly toward a proton that is essentially at rest. If the electron is initially a great distance from the proton, at what distance from the proton is its speed instantaneously equal to twice its initial value (i.e., v f = 2v i ). 14. An infinite sheet of charge has a charge density = 0.12 x 10-6 C/m 2. How far apart are the equipotential surfaces whose potentials differ by 48 V? 18. Compute the escape speed for an electron from the surface of a uniformly charged sphere of radius 1.22 cm and total charge +1.76 x 10-15 C. Neglect gravitational forces. 25. (a) For Fig. 28-34 below, derive an expression for V = V A - V B. (b) Consider the following limiting cases: Does your result reduce to the expected answer when d = 0? When a = 0? When q = 0?
30. Suppose that the electric potential varies along the x axis as shown in the graph of Fig. 28-37 below. Of the intervals shown (ignore the behavior at the endpoints of the intervals), determine the intervals in which E x has (a) its greatest absolute value and (b) its least absolute value. (c) Plot E x versus x. 37. In moving from A to B along an electric field line, the electric field does 3.94 x 10 19 J of work on an electron in the field illustrated in Fig. 28-38 below. What are the following differences in the electric potential (a) V B - V A, (b) V C - V A, and (c) V C - V B? Part B 4. The electric field inside a nonconducting sphere of radius R, containing a uniform charge density, is radially directed and has magnitude E = (qr)/(4 o R 3 ), where q is the total charge in the sphere and r is the distance form the center of the sphere. (a) Find the potential V inside the sphere, taking V = 0 at r = 0. (b) What is the difference in electric potential V between a point on the surface and the center of the sphere? If q > 0, which point is at the higher potential? (c) Show that the potential at a distance r from the center, where r < R, is given by V = q(3r 2 r 2 )/(8 o R 3 ), where the zero of the potential is taken at r = (infinity). Why does this result differ from that of part (a)?
7. A spherical drop of water carrying a charge of +32 x 10-12 C has a potential of 512 V at its surface. (a) What is the radius of the drop? (b) If two such drops of the same charge and radius combine to form a single spherical drop, what is the potential at the surface of the new drop? Set V = 0 at infinity. 8. Figure 28-42 below shows, edge-on, an "infinite" sheet of positive charge density. (a) How much work is done by the electric field of the sheet as a small positive test charge q o is moved from an initial position on the sheet to a final position located at a perpendicular distance z from the sheet? (b) Use the result from (a) to show that the electric potential of an infinite sheet of charge can be written V = V o (/2 o )z, where V o is the potential at the surface of the sheet. 12. A charge per unit length is distributed uniformly along a thin rod of length L. (a) Determine the potential (chosen to be zero at infinity) at point P a distance y from one end of the rod and in line with it (see Fig. 28-45 below). (b) Use the result of (a) to compute the component of the electric field at P in the y direction (along the rod). (c) Determine the component of the electric field at P in a direction perpendicular to the rod. 14. Two identical conducting spheres of radius 15.0 cm are separated by a distance of 10.0 m. What is the charge on each sphere if the potential of one is +1500 V and the other is -1500 V? What assumptions have you made? Take V = 0 at infinity.
Homework #4 Solutions 203-1-1721 Physics 2 for Students of Mechanical Engineering Part A
Part B
Erratum (Corrections to homework) Homework #4: P28-8 qσz (a) The work done by the field is W = F dr = + Fz = + Eqz = +. 2ε 0 (b) Since the work done by the field is σz σz W = q V, then V = and V = V0. 2ε 0 2ε 0 V σ Note that E = = +, z 2ε 0 which confirms that the electric field points in the positive z-direction, as it must.