Exam 1PHY2061 9-29-05 Name:_______________________ ___ Exam 1 Closed book exam. A calculator is allowed, as is one 8.5×11” sheet of paper with your own written notes. Please show all work leading to your answer to receive full credit. Numerical answers should be calculated to 2 significant digits. Exam is worth 100 points, 25% of your total grade. UF Honor Code: “On my honor, I have neither given nor received unauthorized aid in doing this exam.” Sphere: 2244 3.14159273Sr V rπππ=== e=×−16022 1019. C xx yy zzab ab ab⋅= + +ab()()()yz yz xz xz xy xyab ba ab ba ab ba×= − − − + −ab xyz 9201910 N m/C4Kπε==×2. m /s 12 2 208.8542 10 C / N mε−=× c =×30 10812122ˆqqKr=Fr 0q=FE enc0ESqdεΦ= ⋅ =∫ΕΑv 0ρε∇⋅=E V=−∇E 0UVq= CWU d=−∆ = ⋅∫Fs CVd∆=− ⋅∫Es ˆˆˆxyzxyz∂∂∇= + +∂∂∂∂ ()yxzFFFdivxyz∂∂∂∇⋅ = = + +∂∂∂FF VSdV d∇⋅ = ⋅∫∫FFΑv QCV=∆ ()22122QUCVC=∆= eff 1 2CCC=+ eff 1 2111CCC=+ 61 C 10 Cµ−= 61F 10 Fµ−=121 pF 10 F−=191 eV 1.6 10 J−=× 212Kmv= 242bb acxa−± −= 29.8 m/sg =Page 1 of 12PHY2061 9-29-05 Name:_______________________ ___ +2q −7q +4q −5q −3q +3q −2q −3q d−3q −5q +4q −7q +2q 1. [6 points] A central particle of charge −2q is surrounded by a square array of charged particles. The length of the square side is d, and charges are placed at the square corners, at the midpoint of the sides, and midway between the corner and the midpoint for two sides. What are the magnitude and direction of the net electrostatic force on the central particle due to the other particles? Page 2 of 12PHY2061 9-29-05 Name:_______________________ ___ 2. [6 points] Particles 1, with a charge q1 , and 2, with a charge q2, are on the x-axis with particle 1 at x = 2a and particle 2 at x = −a. For the net force on a third charged particle placed at the origin to be zero, what must be q2 in terms of q1 ? Page 3 of 12PHY2061 9-29-05 Name:_______________________ ___ y R x 3. Consider electric charge distributed along a one-dimensional path in the form shown as ¾ of a circle. The circle is centered at the origin with a radius of R, and the linear charge density λ is positive. (a) [6 points] Find the component of the electric field along the x-axis (Ex) at the origin (0,0). (b) [6 points] Find the component of the electric field along the y-axis (Ey) at the origin (0,0). Page 4 of 12PHY2061 9-29-05 Name:_______________________ ___ 4. [8 points] In the figure shown, a small, non-conducting ball of mass and charge 6310 kgm−=×4.8 Cqµ=+ (distributed uniformly through its volume) hangs from an insulating thread that makes an angle θ with a vertical, uniformly charged non-conducting sheet (shown in cross section). The sheet has a surface charge density of . Considering the gravitational force of the ball and assuming that the sheet extends far vertically and into and out of the page, calculate the angle θ. 11 2310 C/mσ−=× Page 5 of 12PHY2061 9-29-05 Name:_______________________ ___ y z x 5. Consider a cube with side length s = 0.5 m and one corner at the origin (0,0,0) as shown. (c) [6 points] What is the total charge enclosed by the cube if the electric field is ()3ˆˆˆ24 6 N/Cyz=+ +Ex y z ? (d) [6 points] What is the electric charge density (C/m3) at the center of the top face at y = 0.5 m if the electric field is the same as in part (a)? Page 6 of 12PHY2061 9-29-05 Name:_______________________ ___ 6. (a) [6 points] A solid non-conducting sphere of radius 0.05 mR=has a uniform charge density constant throughout the volume of the sphere. What is the ratio of the magnitude of the electric field at 30.01 C/mρ=/2rR= to that at rR=? (b) [6 points] Suppose the solid non-conducting sphere in part (a) has a non-uniform charge density (i.e. varies with radius). What is the ratio of the magnitude of the electric field at ()400, where 0.267 C/mrrρρ ρ==/2rR= to that at ? rR= Page 7 of 12PHY2061 9-29-05 Name:_______________________ ___ R3 R1 R2 7. Three charged, concentric conducting shells have radii . The thickness of the conducting shells, while not zero, is considered negligible. The charge on the innermost shell is −15µC, the charge on the middle shell is −20µC, and the charge on the outermost shell is +25µC. 12310 cm, 15 cm, 20 cmRRR=== (a) [6 points] What is the charge on the inner surface of the outermost conducting shell? (b) [6 points] What is the direction and magnitude of the electric field at radius ? 17 cmr = Page 8 of 12PHY2061 9-29-05 Name:_______________________ ___ E+ 8. [8 points] A proton is injected with a velocity of into a region of uniform electric field between two large plates separated by 1m and maintained with an electric potential difference of 30,000 V. The proton travels from lower electric potential to higher on a path perpendicular to the plates. Does the proton reach the far plate, and if not, what is the distance of closest approach to it? The charge of the proton is , and the proton mass is . 6210 m/sv =×191.6 10 Ce−+= ×271.67 10 kgpm−=× v + + + p + + + Page 9 of 12PHY2061 9-29-05 Name:_______________________ ___ 9. [6 points] The electric potential along the x-axis (in kV=103 V ) is plotted versus the value of x (in meters). Evaluate the x-component of the electrical force (in Newtons) on a charge of 4.70 µC located on the x-axis at x= −1.2 m. Page 10 of 12PHY2061 9-29-05 Name:_______________________ ___ A B 10. (a) [6 points] Consider the shown arrangements of capacitors above. Calculate the effective capacitance,, of the network between the terminals A and B given that each of the shown capacitors has a capacitance C = 4 µF. effC (b) [6 points] Consider the shown arrangements of capacitors above. Calculate the effective Consider the above infinite chain of capacitors. Calculate the effective capacitance, , of the network between the terminals A and B given that each of the shown capacitors has a capacitance C = 4 µF. effC A • • • B Page 11
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