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Homework # 6, Electrodynamics 1: 7110
Due Wednesday, Oct 21
Problem 1. A parallel-plate capacitor is stationary in frame K ′ and is oriented in such a
way that the field inside it is parallel to the y ′ axis: E′ = (0, Ey′ , 0). The capacitor is moving
with respect to the laboratory frame K along the x-axis with velocity V . Using Ampere’s
law and Gauss’s law find the magnetic field in the laboratory frame and compare the found
value with the prediction of the Lorentz transformations of the fields. [3 points]
Problem 2. Consider a linear current flowing along the x-axis in frame K. Using Lorentz
transformations of the electromagnetic field, find magnetic and electric fields in the frame
moving with velocity V along the x-axis. Where does the electric field come from? What is
the current in frame K ′ ? What is the charge density? [3 points]
Problem 3. Electric field in the laboratory reference frame K is E = (0, 0, E0 ) and
magnetic field is B = (0, B0 , 0).
a) Find the reference frame K ′ in which electric field is absent E′ = 0. Is this always
possible? If not, what condition needs to be satisfied? [2 points]
b) Same as above but for B′ = 0. [2 points]
Problem 4. Non-relativistic particle with charge q and mass m is placed in electric field
described by the potential ϕ(x, y, z) = k(x2 − y 2 ) (electrostatic lens). Magnetic field is
absent. At the moment of time t = 0 the particle has zero velocity and is located at
r0 = (x0 , y0 , z0 ). Find r(t) for t > 0. [3 points]
Problem 5. Consider a mixed 4-tensor T αβ . Show that T αα is a 4-scalar (remember that
summation is presumed over the repeated indices). Note that T αβ transforms in the same
way as a product of a contravariant and a covariant 4-vectors, Aα Bβ . [2 points]