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4. A solid conducting spherical shell encloses a solid conducting sphere. A charge of -5q is placed on the sphere and a charge of +2q is placed on the shell. The radius of the sphere is a, the inner radius of the shell is b, and the outer radius of the shell is c. These parameters are shown in the figure below. a. Sketch the electric field lines everywhere and show how the charge is distributed over the conductors. b. Calculate the surface charge density on all the surfaces. c. Since the system is spherically symmetric, the electric field will be purely radial E = Er(r)r?. Find the expression for Er(r) for all values of r. d. Can you make the electric field equal to zero between the sphere and the shell simply by changing the charge on the shell? Justify your answer. e. Can you make the electric field equal to zero outside the shell simply by changing the charge on the shell? Justify your answer. 5. Consider an infinitely long cylinder that is concentric with the z-axis. It has a radius of a, and it holds a non-uniform volume charge density of ? = kr, where k is a constant and r is the radial distance from the centre of the cylinder. We would like to calculate the electric field everywhere inside and outside the cylinder. a. Using symmetry arguments, which component(s) of the electric field, Er, E? or Ez, will be zero? b. Draw a cross-section of the cylinder and the draw the Gaussian surface that you would use to calculate the electric field inside the cylinder. c. To find the charge enclosed within the Gaussian surface of (b), you must integrate the charge density. You can do this by considering the cylinder to be made up of an infinite number of thin cylindrical shells. Calculate the differential charge in each shell, while remembering that ? changes with radius, and integrate over the radius from 0 to r to find the total charge within the Gaussian surface. d. Derive expressions for the electric field inside and outside the cylinder. You should get an electric field of kr^2 / (3?0) r? inside the cylinder and ka^3 / (3?0r) r? outside the cylinder. Do these results agree at the surface of the cylinder, where r = a?

4. A solid conducting spherical shell encloses a solid conducting sphere. A charge of -5q is placed on the sphere and a charge of +2q is placed on the shell. The radius of the sphere is a, the inner radius of the shell is b, and the outer radius of the shell is c. These parameters are shown in the figure below.

a. Sketch the electric field lines everywhere and show how the charge is distributed over the conductors.
b. Calculate the surface charge density on all the surfaces.
c. Since the system is spherically symmetric, the electric field will be purely radial E = Er(r)r?. Find the expression for Er(r) for all values of r.
d. Can you make the electric field equal to zero between the sphere and the shell simply by changing the charge on the shell? Justify your answer.
e. Can you make the electric field equal to zero outside the shell simply by changing the charge on the shell? Justify your answer.

5. Consider an infinitely long cylinder that is concentric with the z-axis. It has a radius of a, and it holds a non-uniform volume charge density of ? = kr, where k is a constant and r is the radial distance from the centre of the cylinder. We would like to calculate the electric field everywhere inside and outside the cylinder.
a. Using symmetry arguments, which component(s) of the electric field, Er, E? or Ez, will be zero?
b. Draw a cross-section of the cylinder and the draw the Gaussian surface that you would use to calculate the electric field inside the cylinder.
c. To find the charge enclosed within the Gaussian surface of (b), you must integrate the charge density. You can do this by considering the cylinder to be made up of an infinite number of thin cylindrical shells. Calculate the differential charge in each shell, while remembering that ? changes with radius, and integrate over the radius from 0 to r to find the total charge within the Gaussian surface.
d. Derive expressions for the electric field inside and outside the cylinder. You should get an electric field of kr^2 / (3?0) r? inside the cylinder and ka^3 / (3?0r) r? outside the cylinder. Do these results agree at the surface of the cylinder, where r = a?

4. A solid conducting spherical shell encloses a solid conducting sphere. A charge of -5q is placed on the sphere and a charge of +2q is placed on the shell. The radius of the sphere is a, the inner radius of the shell is b, and the outer radius of the shell is c. These parameters are shown in the figure below.

a. Sketch the electric field lines everywhere and show how the charge is distributed over the conductors.
b. Calculate the surface charge density on all the surfaces.
c. Since the system is spherically symmetric, the electric field will be purely radial E = Er(r)r?. Find the expression for Er(r) for all values of r.
d. Can you make the electric field equal to zero between the sphere and the shell simply by changing the charge on the shell? Justify your answer.
e. Can you make the electric field equal to zero outside the shell simply by changing the charge on the shell? Justify your answer.

5. Consider an infinitely long cylinder that is concentric with the z-axis. It has a radius of a, and it holds a non-uniform volume charge density of ? = kr, where k is a constant and r is the radial distance from the centre of the cylinder. We would like to calculate the electric field everywhere inside and outside the cylinder.
a. Using symmetry arguments, which component(s) of the electric field, Er, E? or Ez, will be zero?
b. Draw a cross-section of the cylinder and the draw the Gaussian surface that you would use to calculate the electric field inside the cylinder.
c. To find the charge enclosed within the Gaussian surface of (b), you must integrate the charge density. You can do this by considering the cylinder to be made up of an infinite number of thin cylindrical shells. Calculate the differential charge in each shell, while remembering that ? changes with radius, and integrate over the radius from 0 to r to find the total charge within the Gaussian surface.
d. Derive expressions for the electric field inside and outside the cylinder. You should get an electric field of kr^2 / (3?0) r? inside the cylinder and ka^3 / (3?0r) r? outside the cylinder. Do these results agree at the surface of the cylinder, where r = a?

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