**Problem 1** [15 points]
Consider a flat conducting sheet lying in the xy-plane at z=0. The sheet is electrically neutral (no net charge) and is carrying a steady uniform (magnetostatic) sheet current **K** = K**ŷ** in the y-direction. It may help conceptually to think of the sheet as having length L, width W, and small thickness d, as in the sketch below, with a uniform current denisty **j**=j**ŷ** flowing through the cross-sectional area dW, so that K=jd. But we are interested in the limits that L and W get infinitely large, while d becomes infintestmally small.

a) Find the electric and magnetic fields **E** and **B** for the above configuration.

b) Consider the above configuration as the inertial rest frame 𝒦 of the sheet. Consider now making a Lorentz transformation to an inertial frame 𝒦' that moves with velocity **v**=v**ŷ** as seen by 𝒦. What are the electric and magnetic fields **E**' and **B**' that are seen in frame 𝒦'?

c) If you did part (b) correctly, you will find that **E**' ≠ 0. Show explicitly that this **E**' results from the presence of a charge density on the sheet, as seen in 𝒦'. Explicitly find this charge density by making a Lorentz transformation of the 4-current from 𝒦 to 𝒦'.

d) Where does this charge come from, given that the sheet is neutral in frame 𝒦? Give a physical answer in terms of simple concepts from relativity, like time dilation or FitzGerald contraction (rather than from the machinery of 4-vectors and trasformation matrices). You only have to give a good explanation, you do not need to do a calculation.

e) Is there a frame 𝒦' in which **B**' = 0? Explain your answer.

**Problem 2** [10 points]
Show that the ordinary acceleration **a**=d**v**/dt of a particle of mass m and charge q, moving at velocity **v** under the influence of electromagnetic fields **E** and **B**, is given by,

**a** = (q/m) (1 - v^{2}/c^{2})^{1/2} [ **E** + **v**×**B** - **v** ( **v**·**E**)/c^{2} ]

Remember to use the correct relativistic form for momentum, **p** = mγ**v**, in Newton's equation **F** = m**a.
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