**Applying Gauss’ Law: Spherical Symmetry**

A point charge causes an electric flux of -750 N . m2/C to pass through a spherical Gaussian surface of 10.0 cm radius centered on the charge. (a) If the radius of the Gaussian surface were doubled how much flux would pass through the surface? (b) What is the value of the point charge 35E. A conducting sphere of radius 10 has an unknown charge. If the electric field 15 ern from the center of the sphere has the magnitude 3.0 X 103 N/C and is directed radially inward what is the net charge on the sphere? ssm 36E. Two charged concentric spheres have radii of 10.0 cm and 15.0 cm. The charge on the inner sphere is 4.00 X 10-8 C, and that on the outer sphere is 2.00 X 10 -8 C. Find the electric field (a) at r = 12.0 cm and (b) at r = 20.0 ern. 37E. In a 1911 paper. Ernest Rutherford said: “In order to some idea of the forces required to deflect an cr particle through a large angle, consider an atom [as] containing a point positive charge Ze at its centre and surrounded by a ‘distribution of negative electricity Uniformly distributed within a sphere of radius R. The electric field E at a distance r from the center for a point inside the atom.

Verify this equation ssm 38E (E = uleo) gives the electric field at points near a charged conducting surface. Apply this equation to a conducting

sphere of radius r and charge q and show that the electric field outside the sphere is the same as the field of a point charge located at the center of the sphere 39P. A proton with speed I’ = 3.00 X lOs m/s orbits just outside

a charged sphere of radius r = 1.00 cm. What is the charge on the sphere A point charge +q is placed at the center of an electrically neutral spherical conducting shell with inner radius a and outer radius b. What charge appears on (a) the inner surface of the shell and (b) the outer surface? What is the net electric field at a distance r from the center of the shell if (c) r < a. (d) b > r > a. and (e) r > b Sketch field lines for those three regions For what is the net electric field due to (f) the central point charge plus the inner surface charge and (g) the outer surface charge? A point charge – q is now placed outside the shell Does this point charge change the charge distribution on (h) the outer surface and (i) the inner surface Sketch the field lines now (j) Is there an electrostatic force on the second point charge (k) Is there a net electrostatic force on the first point charge (I) Does this situation violate Newton’s third law A solid nonconducting sphere of radius R has a nonuniform charge distribution of volume charge density p = PI’IR, where p, is a constant and r is the distance from the center of the sphere. Show (a) that total charge on the sphere is Q = and (b).

gives the magnitude of the electric field inside the sphere.A hydrogen atom can be considered as having a central points proton of positive charge +e and an electron of negative charge that is distributed about the proton according to the volume charge density p = A exp( – 2,./ao). Here A is a constant, ao = 0.53 X 10-10 m is the Bohr radius, and r is the distance from the center of the atom. (a) Using that hydrogen is electrically neutral find A. (b) Then find the electric field produced by the atom at the Bohr radius of radius a and charge +q uniformly distributed throughout its volume is concentric with a spherical conducting shell of inner radius b and outer radius c This shell has a net charge of -q Find expressions for the electric field as a function of the radius r (a) within the sphere (b) between the sphere and the shell (c) inside the shell and (d) outside the shell (r > c). (e) What are the charges on the inner and outer surfaces of the shell a spherical shell of charge with uniform volume charge density p Plot E due to the shell for distances r from the center of the shell ranging from zero to 30 cm. Assume that p = 1.0 X 10-6 C/m’ a = 10 and b = 20 cm.

a nonconducting spherical shell of inner radius a and outer radius b, has a positive volume charge density p = A (within its thickness) where A is a constant and r is the distance’ from the center of the shell In addition a positive point charge q is located at that center What value should A have if the electric field in the shell (a Sr S b) is to be uniform (Hint: The constant

A depends on a but not on b A nonconducting sphere has a uniform volume charge density p Let r be the vector from the center of the sphere to a general point P within the sphere (a) Show that the electric field at P is given by E = pf/3eo. (Note that the result is independent of the radius of the sphere) (b) A spherical cavity is hollowed out of the sphere, as shown in sing superposition concepts show that the electric field at all points within the cavity is uniform and equal to E = pa/ here a is the position vector from the center of the sphere to the center of the cavity (Note that this result is independent of the radius of the sphere and the radius of the cavity.