Homework III, PHY2061 (show work for credit)1.) In the picture, the potential energy at point P is some value, where P lies on the z-axis at -2d on a line connecting the plus charge at +d and the minus charge at –d. Where on the z-axis above the minus charge but below the dashed line (point P’ marked by the x) is the potential energy the same (sign and magnitude) as at point P? – i. e. on the same equipotential line. (This was discussed in class, but rather than some approximate, binomial expansion answer, please give an answer accurate to 3 sig figs. Choose the solution below the dashed line between +q and –q.)2. The 12 charges (6 + and 6 – charges, all equal in magnitude) in the picture below are in a regular array, equally spaced along the horizontal (x-) and the vertical y-axis, with the origin (x=0 and y=0) in the center of the array. Where in the x-y plane is the potential zero? No full credit unless you describe all the possibilities.3.) The text on pages 644 and 645 discusses the exact solution for the potential due to a uniform line of charge, length L, along the z-axis and centered at z=0, at point Pa distance y from the line of charge on its perpendicular bisector at a distance y along the y-axis. The solution, eq. 28-27, is V = (λ/40) ln [ (L/2 + (L2/4+y2)1/2) / (-L/2 + (L2/4+y2)1/2) ]The book then says that in the limit of large (but not infinite) y, that V then looks like it’s from a point charge q, i. e. V=(1/40) λL/y where λL=q. Show, using Taylorseries expansions (see, e. g. , pages A-20 and A-21 in Appendix I) that this is so.4.) A collection of charge produces two equipotential lines like those (black lines) shown in the figure above. (One of the black lines is only partially drawn so as not to crowd the figure.) The red lines drawn tangentially to the two equipotential lines are parallel and are meant to show that along those parts of the equipotential lines the values are essentially constant. Calculate the approximate Electric field (magnitude and direction) at x=y=0, assuming that the two quasi-parallel equipotential lines are 1 meter apart and at an angle of ~50o to the x-axis. Write your answer in vector notation, i. e. E=(…)i +(….)j, where i and j are unit vectors in the x and y directions respectively. (Obviously your gradient differential, e. g in the x-direction, is not in the limit of dx→0.) (Remember that the electric field lines are always perpendicular to the equipotential lines, and
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