Physics 1220                                                                 E from Distributions & Gauss' Law

1. Using the integral form of Coulomb's Law to find
   (a) Find the electric field at a point a distance a from one end of a long thin wire of length L and total charge Q. Examine the limit a > L and show that your result is identical to that of a point charge.
   (b) If the charge distribution was λ(x) = 2Qx/L². Find the electric field. There is no need to evaluate the integral.
   
   

2. Using the integral form of Coulomb's Law, find the electric field at a point a distance a from the centre of a long thin wire of length L and total charge Q. The identity ∫du[u²+v²]^3/2 = u/{v²[u²+v²]^½} + C may be of use. Show that your result reduces to that of a point charge in the limit a >> L. Also show that your answer reduces to E = 2kλ/a in the limit L >> a, the well-known and very useful result for a long thin wire. Note λ = Q/L.
   
   

3. Using the integral form of Coulomb's Law, find the electric field at a point midway between two long thin wires of length L and total charge Q and off to one side a distance h. The distance between the wires is a. The identity ∫du[u²+v²]^-3/2 = u/{v²[u²+v²]^½} + C may be of use. Examine the limit h >> L.
   
   

4. A wire of length 2L is bent in the centre to a right angle. It carries charge density λ. Find an integral expression for the electric field at point P in the upper right corner of the diagram below.
   

5. A wire has been bent into a semicircle of radius R. It has a linear charge density λ. Determine the electric field at point P, at the centre of the circle. The identity S = Rθ and dS = Rdθ may help.
   
6. A thin semicircular wire of radius b has charge Q uniformly distributed along its length. Find an integral expression for the electric field at a point P, a distance a horizontally from its centre.
   

7. A thin semicircular wire of radius b has charge density +λ on the upper half and opposite charge –λ on the lower half. Find the electric field at the centre of curvature.
   

8. Three long thin wires of length L and charges Q, Q, and -Q are arranged to form an equilateral triangle. Use the result of question # 2 to find the electric field at the centre of the triangle.
   
9. The "Gaussian Surface" for an infinitely large charged plate is a pillbox of surface area A and length 2a centred about the origin. The plate has a positive surface charge σ..
   (a) How much charge is contained in the pillbox?
   (b) Symmetry demands that the electric field point in the +x direction on the right side of the plate and -x on the other side. Explain why this is so.
   (c) How much electric field passes through the sides of the pillbox?
   (d) The pillbox has a thickness a on either side of the plate. How do the magnitudes of the electric field compare at x = -a and x = +a?
   (e) Use Gauss's Law to determine the electric field at a.
   (f) Sketch the electric field E(x) as a function of x.
   

10. A very long thick plate has a uniform positive volume charge density give by ρ(x) = ρ₀ for -½d ≤ x ≤ ½d, where ρ₀ is a positive constant, d is the thickness of the plate and x = 0 is the centre of the plate.
    (a) What is the symmetry of the electric field for this object?
    (b) What is the electric field at x = 0? Explain why.
    (c) How much charge is contained in the a "gaussian pillbox" of surface area A and thickness 2a where a < ½d. Use Gauss' Law to find E(a) .
    (d) How much charge is contained in the a "gaussian pillbox" of surface area A and thickness 2a where a > ½d. Use Gauss' Law to find E(a).
    (e) Sketch the electric field for all a.
    

11. The "Gaussian Surface" for cylinders is also a cylinder. It has length L and radius a. Consider an infinitely long wire of uniform charge per unit length λ.
    (a) How much electric field passes through the ends of the "gaussian cylinder". Explain.
    (b) How much charge is contains in the "gaussian cylinder"?
    (c) Use Gauss's Law to determine the electric field at a distance a.
    

12. A very long thin cylindrical shell of radius R carries a negative surface charge density −σ.
    (a) If the radius of the "gaussian cylinder" is smaller than R, i.e. a < R, how much charge is contained by the gaussian surface?
    (b) What, therefore, is the electric field everywhere inside the cylindrical shell?
    (c) How much charge is contained in the "gaussian cylinder" when a > R?
    (d) Use Gauss's Law to determine the electric field at a.
    (e) Sketch the electric field as a function of a.
    

13. A solid sphere of radius R carries a total positive charge Q uniformly distributed thoughout the sphere. The "Gaussian Surface" for a sphere is a sphere of radius a concentric with the sphere.
    (a) What is the electric field at the centre of the sphere? Explain?
    (b) If a < R, how much charge is contained inside the "gaussian sphere"?
    (c) Use Gauss's Law to determine the electric field at the surface of the "gaussian sphere".
    (d) If a > R, how much charge is contained inside the "gaussian sphere"?
    (e) Use Gauss's Law to determine the electric field at the surface of the "gaussian sphere".
    (f) Sketch the electric field.
    

14. A spherical shell of inner radius R and outer radius 2R, has a uniform charge distribution and total charge Q.
    

    (a) Determine the charge inside the "gaussian sphere" for the three regions
    

    1. ) 0 < a < R,
       
    2. ) R < a < 2R,
       
    3. ) 2R < a.
       

    (b) Use Gauss's Law to determine the electric field at the surface of the "gaussian surface" when
    

    1. ) 0 < a < R,
       
    2. ) R < a < 2R,
       
    3. ) 2R < a.
       

    (c) Sketch the electric field for all a.

    

15. Use Gauss's Law to determine the electric field as a function of distance a from the centre of a thin spherical shell of radius R and negative surface charge −σ. Sketch the electric field as a function of a. 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?
    

16. An object consists of two thin spherical shells or radius R and 2R respectively (see diagram below). On the inner shell there is a charge Q and on the outer shell there is a charge –Q. Find and sketch the electric field everywhere.

    

17. An object consists of two thin spherical shells or radius R and 2R respectively (see diagram above). On the inner shell there is charge density σ and on the outer shell there is charge density –σ. What is the charge on each shell? Find and sketch the electric field everywhere. What does the charge on the outside shell have to be for E(r >2R) = 0?
18. A very long cylindrical object consists of a solid core of radius R and a thin shell at radius 2R. The core has linear charge density λ. The shell has linear charge density –λ. What is the charge of a piece of the core of length L? What is the charge of a piece of the shell length L? Find and sketch the electric field everywhere.



19. An object consists of three very large parallel planes of material. The gap between the planes is distance d.  The planes have surface charge densities σ, –σ, and σ from right to left. The planes are shown edge on below. What is the charge of a square of area A of each of these planes? Find and sketch the electric field everywhere. Suppose the planes have surface charge densities σ, 2σ, and 3σ from right to left. Could you still find the electric field? Why or why not.

    

20. A conducting hollow spherical shell carries a net charge of +5Q. A charge of –2Q in placed in the hollow. What will the charge be on each surface of the shell after the charge is put inside?
21. A charge of –2Q in placed in a conducting box which is then placed inside a conducting hollow cylinder. The box was initially uncharged but the cylinder already carried a charge of +2Q. Find the charge on each surface.
    

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Questions? mike.coombes@kwantlen.ca