Physics 2420 In-Class Problems: Gauss' Law

1. Use Gauss' Law to determine the electric field at a distance r from the centre of an infinitely large plate of surface charge Σ. The gaussian surface is a ‘pillbox’.

   

2. A large plate has a charge density give by r (x) = r₀x² for -d/2 £ x £ d/2, where r₀ is a positive constant, d is the thickness of the plate and x = 0 is the centre of the plate. The plate is much longer than it is wide. Use Gauss' Law to find E(x). Sketch the result.

   

3. Use Gauss’ Law to determine the electric field at a distance R to the side of an infinitely long wire of charge per length λ. The gaussian surface is a concentric cylinder.

   

4. A very long thin cylindrical shell of radius R carries a surface charge density Σ. Use Gauss’ Law to determine the electric field as a function of distance r from the centre of the cylinder at a point very far from either end. Sketch the electric field as a function of r.

5. Use Gauss’ Law to determine the electric field as a function of distance a from the centre of a long cylinder of radius R and charge density ρ(r) = Ar². Sketch the electric field as a function of a.

6. A sphere of radius R carries a total charge Q.
   (a) Use Gauss’ Law to determine the electric field as a function of distance a from the centre of the sphere, when a > R. The gaussian surface is a concentric sphere.
   (b) By looking at the electric field, can we distinguish between a sphere with all its charge of the surface and a sphere with the charge smeared throughout the sphere?
   (c) Assuming that the charge density is constant, ρ = Q/V, find the electric field for 0 < a < R.

7. A spherical shell of inner radius R and outer radius 2R, has a uniform charge distribution and total charge Q.
   (a) Determine the charge density ρ(r) for the three regions.
   (b) Find the electric field everywhere.
   (c) Sketch the electric field as a function of r.

   

8. Use Gauss’ Law to determine the electric field as a function of distance a from the centre of a sphere of radius R and charge density ρ(r) = Ar². Find and sketch the electric field as a function of r.

9. A thick conducting spherical shell has out radius R and thickness a. A charge 4Q is placed on the conductor. Then a charge -2Q is placed at the centre of the shell. Find and sketch the electric field as a function of r.

   

10. Determine which of the following electric fields are conservative. If conservative, determine a potential for the field. Determine the charge density r (x,y,z).
    (a) E_x = x+y, E_y = -x+y, and E_z = -2z,
    (b) E_x = 2y, E_y = 2x+3z, and E_z = 3y,
    (c) E_x = x²-z², E_y = 2, and E_z = -2xz.

11. The potential in a certain region of space is given by V = Ar. Determine the charge density r . Determine the total charge in a sphere of radius R centred at r = 0.

Questions? mike.coombes@kwantlen.ca