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Transcript
2004 Electricity and Magnetism I The figure above left shows a hollow, infinite, cylindrical, uncharged conducting shell of inner radius r1, and outer radius r2 . An infinite line charge of linear charge density +λ is parallel to its axis but off center. An enlarged cross section of the cylindrical shell is shown above right. (a) On the cross section above right, i. sketch the electric field lines, if any, in each of regions I, II, and III and ii. use + and - signs to indicate any charge induced on the conductor. (b) In the spaces below, rank the electric potentials at points a, b, c, d, and e from highest to lowest (I = highest potential). If two points are at the same potential, give them the same number. ____Va ____Vb _____Vc _____Vd _____Ve (c) The shell is replaced by another cylindrical shell that has the same dimensions but is nonconducting and carries a uniform volume charge density +ρ. The infinite line charge, still of charge density +λ is located at the center of the shell as shown above. Using Gauss's law, calculate the magnitude of the electric field as a function of the distance r from the center of the shell for each of the following regions. Express your answers in terms of the given quantities and fundamental constants. i. r < r1 ii. r1 < r < r2 iii. r>r2 1992- Electricity and Magnetism I A positive charge distribution exists within a nonconducting spherical region of radius a. The volume charge density p is not uniform but varies with the distance r from the center of the spherical charge distribution, according to the relationship ρ = βr for 0 < r < a, where β is a positive constant, and ρ=0, for r >a. a. Show that the total charge Q in the spherical region of radius a is βπa 4 b. In terms of β, r, a, and fundamental constants, determine the magnitude of the electric field at a point a distance r from the center of the spherical charge distribution for each of the following cases. i. r> a ii. r =a iii. 0 < r <a c. In terms of β, a, and fundamental constants, determine the electric potential at a point a distance r from the center of the spherical charge distribution for each of the following cases i. r= a ii. r =0 1993- Electricity and Magnetism I The solid nonconducting cylinder of radius R shown above is very long. It contains a negative charge evenly distributed throughout the cylinder, with volume charge density ρ. Point P1 is outside the cylinder at a distance r, from its center C and point P2 is inside the cylinder at a distance r 2 from its center C. Both points are in the same plane, which is perpendicular to the axis of the cylinder. a. On the following cross-sectional diagram, draw vectors to indicate the directions of the electric field at points P1 and P2 b. Using Gauss's law, derive expressions for the magnitude of the electric field E in terms of r, R. p, and fundamental constants for the following two cases. i. r > R (outside the cylinder) ii. r < R (inside the cylinder) Another cylinder of the same dimensions, but made of conducting material, carries a total current I parallel to the length of the cylinder, as shown in the diagram above. The current density is uniform throughout the cross-sectional area of the cylinder. Points P, and P2 are in the same positions with respect to the cylinder as they were for the nonconducting cylinder. c. On the following cross-sectional diagram in which the current is out of the plane of the page (toward the reader), draw vectors to indicate the directions of the magnetic field at points P 1, and P2. d. Use Ampere's law to derive an expression for the magnetic field B inside the cylinder in terms of r, R,I, and fundamental constants Homework Tuesday 4-24-2012 A bowling ball of mass 6.0 kg is released from rest from the top of a slanted roof that is 4.0 m long and angled at 30° , as shown above. The ball rolls along the roof without slipping. The rotational inertia of a sphere of mass M and radius R about its center of mass is MR2. (a) On the figure below, draw and label the forces (not components) acting on the ball at their points of application as it rolls along the roof. (b) Calculate the force due to friction acting on the ball as it rolls along the roof. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figure in part (a). (c) Calculate the linear speed of the center of mass of the ball when it reaches the bottom edge of the roof. (d) A wagon containing a box is at rest on the ground below the roof so that the ball falls a vertical distance of 3.0 m and lands and sticks in the center of the box. The total mass of the wagon and the box is 12 kg. Calculate the horizontal speed of the wagon immediately after the ball lands in it.