Download Homework II, due Tuesday, Jan

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 yaxis, 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 P
a 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/40) 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=(1/40) λL/y where λL=q. Show, using
Taylor series 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 vice-versa.)