Basic Concepts>

Bring to the Peer Group Meeting written responses to each of the following. Explain your answers to each question.

1. Define the meaning of each of the symbols that appear in Equation 1: E ⇀ Q [ r ⇀ ] = Q 4 ⁢ π ⁢ ϵ 0 ⁢ r 2 ⁢ r ^ E ⇀ Q [ r ⇀ ] = Q 4 ⁢ π ⁢ ϵ 0 ⁢ r 2 ⁢ r ^ .

2. Clearly distinguish the meanings of source points and field points.

3. Define the meaning of each of the symbols that appear in Equation 2: E ⇀ [ r ⇀ ] = 1 4 ⁢ π ⁢ ϵ 0 ⁢ ∑ n = 1 N ⁢ q n r n 2 ⁢ r ^ n E ⇀ [ r ⇀ ] = 1 4 ⁢ π ⁢ ϵ 0 ⁢ ∑ n = 1 N ⁢ q n r n 2 ⁢ r ^ n .

4. What is the minimum number of point source charges needed to form a charge distribution that would require that at least one of the r ⇀ n ′ r ⇀ n ′ be a 3D vector? Remember that the r ⇀ n ′ r ⇀ n ′ are the location vectors for the source points.

5. Define the meaning of each of the symbols that appear in Equation 3: E ⇀ [ r ⇀ ] = 1 4 ⁢ π ⁢ ϵ 0 ⁢ ∫ V ′ ⁢ ρ ⁢ η ^ η 2 ⁢ ⅆ V ′ E ⇀ [ r ⇀ ] = 1 4 ⁢ π ⁢ ϵ 0 ⁢ ∫ V ′ ⁢ ρ ⁢ η ^ η 2 ⁢ⅆ V ′ .

6. Identify three problems in your textbook such that one each requires charge densities that are 1D, 2D and 3D, respectively. What are the units of each of the three charge densities?

7. State why the calculation in the case of multiple point source charges involves multiple distinct displacement vectors.

8. Perform a dimensional analysis of Equation 3, and demonstrate that the right hand side has the appropriate units for an electric field.

9. Thoughtfully comment on, or critique, the declaration that "Geometry lies at the heart of correctly applying the Coulomb relations."

Preliminary Exercises

Bring your work on these exercises to the Peer Group Meeting.

1. For the dipole example described on Page 2 (the figure is reproduced below), find the electric field everywhere along the positive xx axis, E ⇀ [ { x > 0 , 0 } ] E ⇀ [ { x > 0 , 0 } ] ; that is, the field point has coordinates { x > 0 , 0 } {x>0,0}.

Figure 1: Figure 3. Coulomb vectors for a dipole aligned along the zz axis with a field point on the xx axis.

Coulomb Vectors for a Dipole

2. Consider the solution obtained for E ⇀ [ { x , 0 } ] E ⇀ [ { x , 0 } ] found above. Is this function even or odd? Explain, including the meaning of the terms even and odd applied to a function. Plot the function as a graph.

3. Select from the end-of-chapter problems in your textbook an exercise that is a straightforward application of Equation 1, and work it out.

4. Select from the end-of-chapter problems in your textbook an exercise that is a straightforward application of Equation 2, and work it out.

5. Select from the end-of-chapter problems in your textbook an exercise that is a straightforward application of Equation 3, and work it out. A 1D or 2D exercise is OK, so that, instead of ρ ρ, you use either σ σ or λ λ as the charge density.

6. In the worked out example of Page 3, the needed value of η ^ η 2 η ^ η 2 was determined by independently finding values of η ^ η ^ and η 2 η 2 . Explain how the same result could have been obtained more directly using the identity η ^ η 2 = η ⇀ η 3 η ^ η 2 = η ⇀ η 3 .

7. Read the problems to be addressed in the Peer Group meeting, and plan strategies for solving them.