1) (10 Points) On its mission to explore the UNIVERSE, the USS Enterprise ( m = = 100,000 [kg ] ) discovers an infinite length rod in an otherwise empty region of space. The captain decides to send out a probe to study the properties of the rod. The Enterprise "parks" 200 [km] from the rod, holding itself at rest relative to the rod by applying a constant thrust force away from the rod with a magnitude of 1 [MN] (1 [MN] = 1,000,000 Newtons). It then fires out a "probe" which is effectively a bowling ball with a mass mg=20[kg ] and charge 9:=+5[uC] . The "probe" moves about the rod in a circular orbit with speed 15,000 [m/s]. a) Draw a FBD of the (uncharged) Enterprise at rest near the rod. b) Use Gauss' Law for Gravity to find an expression for the gravitational field due to the infinite length rod a distance r away in terms of the linear mass density of the rod ,, . GL for gravity is of g. dA=-4R G mene where Mene is the mass enclosed by your chosen Gaussian surface (GS). HINT: You must draw your GS and go through the steps. c) Find the linear mass density A,,, of the rod. d) Draw a FBD of the orbiting bowling ball "probe" e) Find the linear charge density ~, 2) (10 Points) A 2D annulus (thick ring) has an inner radius R, and outer radius R. , and charge Q non-uniformly distributed over its surface. The 2D charge density varies with radius r by n (r)=Cr* for R,Er ER, and n=0 for all other r. C is a constant. Answer the following in terms of the variables given above. Note: Gauss' Law will not be useful here. a) Find an expression for C such that the total charge of the annulus is Q. Include the SI units for C next to your answer. Do the units make sense? b) Draw a clear picture and use it to set up the integral to calculate the E-field at a point on the axis of the annulus (this axis is perpendicular to the plane of the annulus) a distance z from the center of the annulus. * ** You must complete all the steps short of actually computing the integral (i.e. your final answer must be an integral with only ONE variable of integration and all other variables constant.) Show that your answer has the correct SI units for the electric field. 3) (10 Points) Four point charges are held fixed in space on the corners of a rectangle with a length of 20 [cm] (in the horizontal direction) and a width of 10 [cm] (in the vertical direction). Starting with the top left corner and going clockwise, the charges are q,=+10 [nc], 92=-10[nC] , q;=-5[nC] , and 94=+8[nc] . a) Find the magnitude and direction of the electric force on charge 94 b) Find the magnitude and direction of the electric field at the midpoint between 9, and 94 c) Find the magnitude and direction of the electric field at the center of the rectangle