I would like some help to solve this problem and also if you can provide explanations.

Introduction to Electromagnetism and Optics 1402 Problem Set 10, due 4/29/24 11:59 pm FEach of these problems was basically discussed in class. See if you can do them without looking at the lecture notes. If not, look at the notes and make sure you understand them before writing down your selutions. 1. Take a look at Figure 1. It depicts two different circular motions (of a point charge of charge and mass m) in a uniform magnetic field. One of them is allowed, and the other is actually not (meaning it's inconsistent with the laws of mechanics and electromagnetism as we know them). Tell me which is which, assuming g > (, and explain your reasoning. How would your answer change if 0 and B > 0. Figure 1: Both Figure la and Figure 1b depict a uniform magnetic field B pointing into the page (B > 0), and a charged particle of charge and mass m in circular motion with speed v and radius r. In Figure la, the motion is counter-clockwise. In Figure 1b, it's clockwise. 2. Take a look at Figure 2. It depicts two cases (A and B): a circular loop with current following in indicated directions (each presented with both a 3D view, and a cross-sectional view). Draw the magnetic field lines (including direction) for the two cases. I ask you to draw the magnetic field lines both in the 3D view and the cross-sectional view, for each of the two cases. The precise magnetic field in both cases, while qualitatively known (as I ask you to draw), is actually not easy to compute quantitatively. But it's not hard to figure out the precise magnetic field at the center of the loop, using Biot-Savart Law. Write down an expression for the magnetic field at the loop center (for both case A and case B), in terms of the current I and radius r. A parenthetical remark: the Biot-Savart Law can actually be used to compute the magnetic field everywhere, but it requires doing a tricky integral. Care A Can B L L L > e AR Covtr- = . o . > fairms o S [ Toeo T e e o ra* Figure 2: Case A and Case B each represents a circular wire loop with current I flowing in the indicated direction. For each case, I show both a \"3D\" view and a \"cross-section\" view. 3. Take a look at Figure 3. It depicts two long (practically infinite), parallel, straight wires, with wire 1 carrying current I; and wire 2 carrying current I respectively. First, use Ampere's Law to figure out the magnetic field sourced by I; at the location of wire 2. Then, find the resulting magnetic force on wire 2. Make sure you give the direction of the magnetic force. Second, use Ampere's Law to figure out the magnetic field sourced by I5 at the location of wire 1. Then, find the resulting magnetic force on wire 1. Make sure you give the direction of the magnetic force. Do you find the two forces obey Newton's third law: are they equal and opposite? everlf={ecfa JiEgw I, wte prye Ty e prge of Figure 3: Two straight, infinitely long, parallel wires are depicted, with current I; and I respec- tively. The wires are separated by a distance d. On the left is the \"3D\" view, and on the right is the cross-section view with the wires perpendicular to the page. 4. Take a look at Figure 4 on the next page. It depicts four cases (A, B, C and D): each with a uniform magnetic field (but confined to the right half of the diagram), and a wire loop being pushed into or out of the magnetic field. Draw the direction of the expected electric field, which is also the direction of the expected current, for each case. Let us focus on case A. Write down an expression for EMF that is generated around the wire loop (the line integral of the electric field), in terms of the speed v, the magnetic field (magnitude) B and the length L. Assuming the wire has resistance R, write down an expression for the expected current I as a result, in terms of R, v, B an L. Lastly, write down an expression for the resulting magnetic force on the wire. To keep the wire loop moving at constant velocity, whoever is pushing the wire needs to push it with exactly the opposite of this force. This means the person is doing work to push the wire loop into the magnetic field. What is the work done (which is energy) per unit time, which we usually call power? Remember energy is always conserved. The energy input into the system has to come out somehow. Where does this energy go? 5. This problem is optional and will carry no credit. Here, I want to show you how a wave equation works. It's pretty marvelous! Consider the following equation: 82f o czazf =0. ot? Ox? Here, f is a function of both time and space z, and c is a constant which we will later see is the wave speed. Remember how partial derivatives work: 3'% means a time derivative holding z fixed, and 3'9; means a spatial derivative holding fixed. Show that the following is a solution of the wave equation: (1) f = sin (kz wt) (2) where k and w are constants, satisfying the relation w = ke. Draw what f looks like as a function of = at time = 0. You will see that f looks like a wave with crest and trough, and the distance Case A Case B X X X X X X X X X X X X X X X X x X X X no magnetic uniform B no magnetic field on this uniform B into page field in this side side into page Case C Case D no magnetic uniform B no magnetic field D this uniform B side out of page field s this side out of page Figure 4: Case A, B, C, D each represents a rectangular wire loop moving towards or away from a region of uniform magnetic field B (on the right half of each diagram). The crosses and dots indicate the direction of the magnetic field. The left half of each diagram has no magnetic field. between two successive crests is 27/k (this is called the wavelength). Next, imagine a short time later, let's pick say t = 0.1/w. Convince yourself f looks exactly the same as what you drew for t = 0, except that the wave pattern has shifted slightly to the right. We say that your wave propagates to the right. Convince yourself the distance by which it propagates in that amount of time t = 0.1/w is 0.1/k. Thus, dividing distance by time, we have w/k, which is why it's correct to identify c as the wave speed. A few more comments. (1) The solution in Eq. (2) can of course by multiplied by an arbitrary constant-that determines the wave amplitude. (2) Here, I single out a to make the discussion simple. In general, we can have a wave f which is a function of t and x, y, z. And the wave can propagate in any direction you wish. (3) In electromagnetism, it can be shown that in vacuum (no charges and currents), the electric and magnetic fields each satisfy a wave equation, i.e. replace f by E or B. The wave equation follows directly from the Maxwell equations