Physics 2 for Students of Mechanical Engineering

Transcription

1 Homework # 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 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 x C. Neglect gravitational forces. 25. (a) For Fig 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?

2 30. Suppose that the electric potential varies along the x axis as shown in the graph of Fig 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 J of work on an electron in the field illustrated in Fig 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)?

3 7. A spherical drop of water carrying a charge of +32 x 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 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 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 V and the other is V? What assumptions have you made? Take V = 0 at infinity.

4 Homework #4 Solutions Physics 2 for Students of Mechanical Engineering Part A

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6 Part B

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8 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.