Tutorial Divergence. (ii) Explain why four of these integrals are zero, and calculate the other two.

Transcription

1 (1) Below is a graphical representation of a vector field v with a z-component equal to zero. (a) Draw a box somewhere inside this vector field. The box is 3-dimensional. To make things easy, it is a good idea to let the edges of the box run parallel to the arrows of the vector field,, so the box will look like a rectangle in the xy-plane. (b) We will now calculate the net flux v da through the surface of this box, S knowing that v = cxx with c a constant and S the surface that encloses the volume of the box. Let the lower left corner of your box have the coordinates (x, y, z). Call Δy the height, Δx the length and Δz the width of the box. (i) Write v da as a sum of 6 integrals. S (ii) Explain why four of these integrals are zero, and calculate the other two. Developed by KU Leuven / DCU / University of St Andrews 1

2 (iii) Show that the total net flux is equal to v da S = cδxδyδz. (c) Write down the divergence theorem, and use it to explain in words why the divergence of the field should be non-zero somewhere inside the box. (d) Calculate the divergence at an arbitrary spot inside the box using 1 v = lim v da V 0 V S where V represents the volume of the box that is enclosed by S. This volume shrinks down to zero at the location where the divergence is calculated. (e) Confirm your result from part (d) with a direct calculation in Cartesian coordinates of the divergence of the field at an arbitrary point inside the box. (f) Explain in your own words what the connection between part (d) and part (e) is. Developed by KU Leuven / DCU / University of St Andrews 2

3 (g) In the previous calculations we showed how you can see whether the divergence of a field is zero or not at a certain location by checking the balance between the inward flow and outward flow through a box, while letting the box become very small. Suppose the vector field is changed to the one that is given below. Where is the divergence of this field zero and where is it non-zero? Show it by using the box mechanism and check your answer with a calculation! (h) The divergence is often described as a measure of the magnitude of the source or sink of a vector field at a certain location. Explain this using the previous two examples. (2) In the vector fields that are discussed above, the divergence is a constant and therefore the same at every location in the field. However, it is also possible that the strength of the source or sink varies over the field. Give an example of such a field, and make a sketch of the field vectors. Developed by KU Leuven / DCU / University of St Andrews 3

4 (3) The vector fields shown in the diagrams below all have cylindrical symmetry. Each diagram shows the field in the xy-plane, and the z-component of each field is zero. (a) Below is a graphical representation of a vector field with a magnitude that increases with increasing distance from the origin. (i) In the vector field diagram a box (it is 3D!) is drawn. Due to symmetry reasons, we choose not to use a rectangular box in this case. Where (if anywhere) would you expect the divergence of the field to be nonzero? (ii) This field is v = cs = css with c a positive constant. Calculate the divergence of the field at an arbitrary location. (iii) Compare the answer in part (ii) to your expectations from part (i). If they are inconsistent, reconsider your reasoning. Developed by KU Leuven / DCU / University of St Andrews 4

5 (b) The following vector field decreases in magnitude when the distance to the origin increases. (i) At first sight, where (if anywhere) would you expect the divergence to be nonzero in the field? Why? (ii) This field is v = c s with c a positive constant. Calculate the divergence of s the field at an arbitrary location. (iii) What happens at the origin? (Hint: think about delta distributions) Developed by KU Leuven / DCU / University of St Andrews 5

6 (iv) Compare the answer in part (ii) to your expectations from part (i). If they are inconsistent, reconsider your reasoning. (v) Can you think of a physical field that behaves like this? Give at least one example. (c) Sometimes it is hard to draw and interpret 2D figures of 3D fields. For example, imagine that the diagram on the previous page is actually an intersection between the xy-plane and the spherically symmetrical v = c r field. In this case the zcomponent of the field is nonzero, but the graphical representation of the r intersection with the xy-plane looks exactly the same as in the previous question. (i) Calculate the divergence of such a field. Explain the result. (ii) What spherically symmetric vector field can you think of that has a zero divergence everywhere except at the origin? Prove your answer with a calculation. Give at least one example of a physical field that shows this behavior. Developed by KU Leuven / DCU / University of St Andrews 6