TASK 2, DIVERGENCE, CURL, ELECTRIC FIELD

Problem 1:
Test the divergence theorem for the function v = (xy) X+ (2yz) y+ (3zx) Z. Take as your volume the cube shown in Fig. 1., with sides of length 2.

Problem 2:
Test Stokes’ theorem for the function v = (xy) X+ (2yz) Y+ (3zx) Z, using the triangular shaded area of Fig. 2.

Problem 3
(a) Twelve equal charges, q, are situated at the comers of a regular 12-sided polygon (for instance, one on each numeral of a clock face). What is the net force on a test charge Q at the center?
(b) Suppose one of the 12 q’s is removed (the one at “6 o’clock”). What is the force on Q? Explain your reasoning carefully.
(c) Now 13 equal charges, q, are placed at the comers of a regular 13-sided polygon. What is the force on a test charge Q at the center?
(d) If one of the 13 q’s is removed, what is the force on Q? Explain your reasoning.
Problem 4
(a) Find the electric field (magnitude and direction) a distance z above the midpoint between two equal charges, q, a distance d apart (Fig. 4). Check that your result is consistent with what you’d expect when z » d.
(b) Repeat part (a), only this time make the right-hand charge -q instead of +q.

Figure 4

Problem 5
Find the electric field a distance z above one end of a straight line segment of length L (Fig. 5), which carries a uniform line charge A. Check that your formula is consistent with what you would expect for the case z » L.

Figure 5                                                 Figure 6                                           Figure 7
Problem 6
Find the electric field a distance Z above the center of a square loop (side a) carrying unifonn line charge A (Fig. 6).

Problem 7
Find the electric field a distance Z above the center of a circular loop of radius r (Fig. 7), which carries a uniform line charge A.

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