3 01/28 F

1. 1st Ed: p78, q60, 2nd Ed: p92, q60. (20 points) Also find two scalar equations that describe the line through P that crosses the surface normally at P.

   Find the unit normal $\vec n$ to the surface at P. Now assume that the surface is reflective, satisfying Snell's law. An incoming light beam parallel to the $x$-axis hits the surface at P. Find a vector equation that describes the path of the reflected beam.

   Hint: let $\vec v$ be a vector along the light ray. The component of $\vec v$ in the direction of $\vec n$ is $\vec n\cdot\vec v$. The component vector in the direction of $\vec n$ is defined as $\vec v_1=\vec n(\vec n\cdot\vec v)$. Sketch this vector along with vector $\vec n$. In which direction is the remainder $\vec v_2 =\vec v - \vec v_1$? Now figure out what happens to $\vec v_1$ and $\vec v_2$ during the reflection. Take it from there.

2. 1st Ed: p80, q87, 2nd Ed: p93, q87. (20 points) Compare with a point sink in which
   
   $$\vec v = -\frac{x{\hat\imath}+y{\hat\imath}}{x^2+y^2}$$
   
   
   For each flow, compute the divergence, draw streamlines, and figure out how much fluid passes through a circle of arbitrary radius $r$. (Since the velocity is radial, the fluid flow through a circle is the magnitude of the velocity times the circumference of the circle.) Based on the results, explain where all the fluid that enters the unit circle disappears. In particular, at an arbitrary point $r,\theta$, what is the amount of fluid disappearing per unit area? So, what do you think of the value of the divergence of the point sink at the origin?

3. 1st Ed: p80, q102, 2nd Ed: p94, q102.

4. 1st Ed: p81, q107, 2nd Ed: p94, q107. (20 points). You need to show that any solution $\vec E,\vec H$ of Maxwell’s equations is given by scalar and vector potentials $\phi,\vec A$ as shown. Hints:

   Recall that if the divergence of a vector is zero, the vector is the curl of some other vector $\vec A_0$. (Actually, I forgot to tell you that, but you know it now.)

   Also, you can certainly define $\vec E_\phi$ by setting
   

   $$\vec E = - \frac{1}{c} \frac{\partial \vec A_0}{\partial t} + \vec E_\phi$$
   
   
   but it is not automatic that $\vec E_\phi$ is minus the gradient of some scalar $\phi_0$. That is for you to show.

   Unfortunately, $\vec A_0$ and $\phi_0$ are not unique and do not normally satisfy (1) in the book. The potentials you need are of the form
   

   $$\vec A = \vec A_0 + \nabla \psi \qquad \phi = \phi_0 - \frac{1}{c} \frac{\partial \psi}{\partial t}$$
   
   
   Show that in those terms,
   
   $$\vec E = - \frac{1}{c} \frac{\partial \vec A}{\partial t} -\nabla\phi \qquad \vec H = \nabla \times \vec A$$
   
   
   regardless of what you take for $\psi$. (That is the famous gauge property of the electromagnetic field.) The way that you want to take $\psi$ is so that equation (1) in the book is satisfied. Show that this leads to a partial differential equation for $\psi$.

   Now substitute into the four Maxwell equations and so find the requirements that $\vec A$ and $\phi$ must satisfy.

   Show directly from Maxwell's first and last equation that the charge density must not vary in time. (That is because the current density in the last equation was left out. There should be a $4\pi\vec j/c$ in the last equation, as well as in (3).)

5. 1st Ed: p102, q32, 2nd Ed: p122, q32.

6. 1st Ed: p103, q44, 2nd Ed: p123, q44. Do it without using Stokes.