1. (a) Explain space-like and time-like 4-vectors with examples. (b) Calculate the 4-velocity of a particle...

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1. (a) Explain space-like and time-like 4-vectors with examples. (b) Calculate the 4-velocity of a particle in an inertial frame where its co-ordinates are given as, x(t) = at +b cos wt, y(t) = 0 and z(t) = b sin wt. (c) An observer on earth records two events, P at t = 0 on earth and Q at t = 1 year at Proxima Centauri. The Proxima Centauri is located at 4.25 light-years away from the earth. Find the time difference between the two events according to an observer moving with a velocity 0.989 c from Proxima Centauri to earth. 4+3+3 2. The electric field of an EM wave in the laboratory frame is E = Eo cos[(xy)-wt] 2. Consider an inertial frame which is moving with uniform velocity v along the negative y axis in the laboratory frame. (a) Write down the 4-wave vector associated with the EM wave in the laboratory frame. (b) Find the frequency of the EM wave in the moving frame. (c) What is the direction of propagation of the EM wave in the moving frame. 3+4+3 3. Consider an electromagnetic field E = 2 x 104 V/m and B = 3 x 10-5 Tesla 2 in an inertial frame S. Consider another inertial frame S' in which the frame S appears to move with speed v = 0.999 c along the y axis. (a) Write F and F in the S frame. (b.) What is the boost matrix A from S to S"? (c) Calculate E and B' in the S' frame. 4. (a) Reformulate the Lorentz force law in tensor notation. (2+2)+2+(2+2) 1. (a) Explain space-like and time-like 4-vectors with examples. (b) Calculate the 4-velocity of a particle in an inertial frame where its co-ordinates are given as, x(t) = at +b cos wt, y(t) = 0 and z(t) = b sin wt. (c) An observer on earth records two events, P at t = 0 on earth and Q at t = 1 year at Proxima Centauri. The Proxima Centauri is located at 4.25 light-years away from the earth. Find the time difference between the two events according to an observer moving with a velocity 0.989 c from Proxima Centauri to earth. 4+3+3 2. The electric field of an EM wave in the laboratory frame is E = Eo cos[(xy)-wt] 2. Consider an inertial frame which is moving with uniform velocity v along the negative y axis in the laboratory frame. (a) Write down the 4-wave vector associated with the EM wave in the laboratory frame. (b) Find the frequency of the EM wave in the moving frame. (c) What is the direction of propagation of the EM wave in the moving frame. 3+4+3 3. Consider an electromagnetic field E = 2 x 104 V/m and B = 3 x 10-5 Tesla 2 in an inertial frame S. Consider another inertial frame S' in which the frame S appears to move with speed v = 0.999 c along the y axis. (a) Write F and F in the S frame. (b.) What is the boost matrix A from S to S"? (c) Calculate E and B' in the S' frame. 4. (a) Reformulate the Lorentz force law in tensor notation. (2+2)+2+(2+2)