Chapter 3 ELECTRIC FLUX DENSITY, GAUSS LAW, AND DIVERGENCE(1) An empty metal point can is placed on a marble table, the lid is removed, and both parts are discharged (honorably) by touching them to ground. An insulating nylon thread is glued to the center of the lid, and a penny, a nickel and a dime are glued to the thread so that they are not touching each other. The penny is given a charge of +5nC, and the nickel and dime are discharged. The assembly is lowered into the can so that the coins hang clear of all walls, and the lid is secured. The outside of the can is again touched momentarily to ground. The device is carefully disassembled with insulating gloves and tools.(a) What charges are found on each of the five metallic pieces?

(b) If the penny had been given a charge of +5nC, the dime a charge of -2nC, and the nickel a charge of -1 nC, what would the final charge arrangement have been?

(3) A non-uniform surface chare density of nC/m2 lies in the plane z = 2 wherever < 5; s = 0 for > 5.

(a) How much electric flux leave in the circular region < 5, z = 2?

(b) How much electric flux crosses the z = 0 plane in the az direction?

(c) How much electric flux leaves the cylinder = 3 in the a direction?

(5) A point charge of 6C in located at the origin, a uniform line charge density of 180nC/m lies along the x axis, and a uniform sheet of charge equal to 25nC/m2 lies in the z = 0 plane.

(a) Find D at A(0,0,4)

(b) Find D at B(1,2,4)

(c) Calculate the total electric flux leaving the surface of 4m radius centered at the origin.

(7) Uniform line charges of 20nC/m each lie in the z = 0 plane of x = 0, 1, 2,, 5.

(a) Find the total electric flux leaving the spherical surface r=2.5

(b) Find D at P (0,2.5,4)

(9) The spherical region, 0 < r < 10 cm, contains a uniform volume charge density v = 4C/m3.

(a) Find Qtot, 0 < r < 10 cm

(b) Find Dr, 0 < r < 10 cm

(c) The non-uniform volume charge density v = nC/m3, exists for 10 cm < r < ro so that the total charge, o < r < ro, is zero.

(11) Let v = 0 for < 1 cm and also for > 3cm. In the region 1 < < 3cm, v = C/m3. Find D everywhere.

(13) Cylindrical surface at = 2, 4 and 6m carry uniform charge densities of 20 nC/m2, -4nC/m2, and s0, respectively.

(a) Find D at = 1, 3, and 5m

(b) Determine s0 such that D = 0 at = 7m.

(15) Uniform surface charge densinty of 100, -60, and 50 C/m2 lie in x = 0, y = 0, and z = 0 planes, respectively. Find at point 2.5 m from every coordinate plane in each of the eight octants.

(17) Let C/m2

(a) Use Gauss law to determine the charge enclosed in the cubical region,

(b) Use Eq.(8) to obtain an approximate value for the above charge. Evaluate the derivatives of the center of the volume.

(c) Show that your answers agree in the limit as a becomes very small.

(18) (19) Let C/m2 and find the increment amount of charge in a volume of 10-10 m3 located at:

(a) (0,0,0)

(b) (4,2,-3)

(c) (4,y,-3)

(d) At what location in the cubical region should the volume be located to contain a maximum charge?

(21) Calculate the divergence of G at P(2,-3,4) if G = :

(a)

(b)

(c)

(d)

(23) The electric flux density C/m3 in the region m.

(a) Find the v at r = 0.4m

(b) Find Dr at r = 0.4m

(c) What total charge is contained in the region m

(d) If v = 0 for r > 1 m, find Dv for r > 1m.

(25) If C/m2, find

(a)

(b) at P(x,y,z)

(c) The total charge lying withing the region .

(d) The total charge lying without finding v first.

(27) Within the cylindrical region m, the electric flux density is given as C/m2

(a) What is the volume charge density at = 3m?

(b) What is the electric flux density at = 3?

(c) How much electric flux leaves the cylinder, = 3, ?

(d) How much charge is contained within the cylinder, = 3, ?

(29) If D = 0.1rar C/m2 for m, and :

(a) Find , v at r=0.2 and 0.5m

(b) What point charge could be placed at the origin to cause D to be zero for ?

(31) Given the field C/m2, find the total charge lying within the volume, by two different methods.

(33) Let and find the value of over the surface of the shphere r = 1.