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# Special Relativity and Schwarzschild Radius

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1. A particle is measured to have a kinetic energy that is four times its rest as energy. How fast is the particle moving? Express the speed as a fraction of c.

2. Write down the transformations that relate the electric field E and the magnetic field B in two different inertial frames. Use your result to show that E*B is the same for all inertial observers. You may choose the motion to be in the x-direction if you wish.

3. Define the Schwarzschild radius of a black hole and give a brief description of its significance.

#### Solution Preview

1: Kinetic energy
K = 4*mo*c^2 = mo*c^2 [{1/sqrt(1- u^2/c^2)} - 1]
=> 4 = [{1/sqrt(1- u^2/c^2)} - 1]
=> 5 = 1/sqrt(1- u^2/c^2)
=> 1/25 = 1- u^2/c^2
=> u^2/c^2 = 1 - 1/25 = 24/25

2: Let us assume frame S' moves along x-x' (common to S and S') with velocity v with respect to frame S.

Therefore,
E'x = Ex
E'y = g*(Ey - v*Bz)
E'z = g*(Ez + v*By)

B'x = Bx
B'y = g*(By + ...

#### Solution Summary

This solution contains three answers with explanation and calculations. 300 words.

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