Gauss's Law

• Dot Product (for finding the components of vector field perpendicular to surface)
• Cross Product (for finding unit vector normal to surface)
• Electric Field as vector field (for calculating electric flux)
• Surface Element in Rectangular and Curvilinear Coordinates (finding unit vector normal to surface)
• Scalar Surface Integrals (for finding total flux)
• Volume Element in Rectangular and Curvilinear Coordinates (for finding total enclosed charge for Gauss' Law)
• Volume Integrals (for finding total enclosed charge for Gauss' Law)
• Charge Density (for finding total enclosed charge for Gauss' Law)
• Total Charge (for Gauss' Law)
• Partial Derivatives (for derivation of the Divergence Theorem)
• Limit Definition of Derivative (for derivation of the Divergence Theorem)

Related Prior Homework

• CrossTriangle: students calculate the unit vector normal to a slanted surface

Optional:

• Theta Functions (for special charge distributions)
• Delta Functions (for special charge distributions)

1. (GaussLaw) This problem is a straightforward follow-up to the Gauss's Law activity. Part (b) can be included as an introduction to the concept and usefulness of theta functions.

   A positively charged dielectric cylindrical shell of inner radius $a$ and outer radius $b$ has a cylindrically symmetric internal charge density $$\rho = 3\,\alpha \; \sin\left(\frac{\pi(s-a)}{b-a}\right)$$ where $\alpha$ is a constant with appropriate dimensions.

   1. Sketch the charge density and find the total charge on the shell.

   2. Write the volume charge density everywhere in space as a single function.

   3. Use Gauss's Law and symmetry arguments to find the electric field in each of the regions given below:

      (i) $s < a$

      (ii) $a < s < b$

      (iii) $s > b$

   4. Sketch the $s$-component of the electric field as a function of $s$.

2. (GaussLawLimit) Take the limit in the previous problem so that the cylindrical shell becomes infinitely thin. The results of this problem can be used as a nice introduction to electrostatic boundary conditions. (See Unit Boundary Conditions). Part (b) can be used as an introduction to the concept and usefulness of delta functions.

   Referring to the charge distribution in the Gauss's Law problem which you have solved above, take the limit as $a\to b$ so that the shell becomes infinitely thin, but keeping the total charge on a unit length of the cylinder constant. Redo each part of the previous problem for this situation.

   1. Find the surface charge density on the shell.

   2. Write the volume charge density everywhere in space as a single function.

      Be careful: Integrate your charge density to get the total charge as a check.

   3. Use Gauss's Law and symmetry arguments to find the electric field at each region given below:

      (i) $s < b$

      (ii) $s > b$

   4. Sketch the $s$-component of the electric field as a function of $s$.

   5. Compare the surface charge density on the shell to the discontinuity in the $s$-component of the electric field.

3. (GaussLawLimitChallenge–Challenge) In this challenging problem you take the limit of previous problem as the shell shrinks to a point source. This problem should probably be reserved for advanced students.

   Take the limits of the shell in the previous problem as $a\to b$ and then $b\to0$, so that the shell becomes a charged line, but keeping the total charge on a unit length of the cylinder constant.

   1. Find the charge density on the line.

   2. Give a formula for the charge density everywhere in space.

      Be careful: Integrate your charge density to get the total charge as a check.

   3. Use Gauss's Law and symmetry arguments to find the electric field for $r>0$.

4. (Symmetry)

   For each of the following situations, can you use Gauss' Law to find the electric field at an arbitrary point, $P$, located outside of the charge distribution? \begin{itemize}

5. If no, explain why not.
6. If yes, \begin{itemize}
7. draw the Gaussian surface you would use and describe why you chose that shape and orientation (\emph{i.e.} make explicit symmetry arguments), and
8. use Gauss' Law to find the electric field at point $P$.