Field transformation laws - Relativity

[erisedk]

Homework Statement

The electric and magnetic fields of a 1 Coulomb charge Q are measured by a pair of field measuring instruments. From the perspective of observers in frame O, the charge is at rest at the origin and one of the field-measuring devices is also at rest, with position (x,y,z) = (0,1,0). Observers in O see the second field-measuring device moving at high speed, with velocity v = βc (x direction) (it travels along the line y = 1)

(a) At time t = 0 this device passes very close to the stationary device at (x,y,z) = (0,1,0). The readings on the dials of the moving device were photographed by observers in O as it was illuminated by the lights from the stationary device's panel lights.

1. Calculate the components of the electric and magnetic field vectors measured by the stationary device at the time the photograph was taken.

2. Calculate the components of the electric and magnetic field vectors measured by the moving device.

3. According to observers in the rest frame of the moving device, how far away is the charge Q?

Homework Equations

The Attempt at a Solution

1. Electric field vector at stationary device = kQ/r² (j) = (9 × 10⁹ × 1)/1² = 9 × 10⁹ V/m

Magnetic field vector at stationary device = 0 because there is no relative motion between the point charge and the stationary device.

2. I don't understand how to do this at all. We're supposed to used this large matrix (EM strength tensor) I think, but I don't really understand how to do that.

If I were to do it without relativity, I'd just treat Q as a line of charge (or a current carrying wire) and solve, but that's definitely wrong.

3. Wouldn't it keep changing as a function of time?
d = $\sqrt{1^2 + v^2t^2}$ where v = βc

Please help.

 

[mark726]

1. Your calculation for the electric field vector at the stationary device is correct. However, the magnetic field vector is not necessarily zero. Since the charge Q is moving, there will be a magnetic field at the location of the stationary device. The magnetic field vector can be calculated using the Biot-Savart law.

2. To calculate the components of the electric and magnetic field vectors measured by the moving device, you will need to use the Lorentz transformation equations. These equations relate the measurements made in one frame of reference (in this case, frame O) to the measurements made in a different frame of reference (the frame of the moving device). The EM strength tensor you mentioned is a mathematical tool that is used to represent the electric and magnetic fields in relativity. You can use this tensor to calculate the components of the fields in the moving frame of reference.

3. Yes, the distance between the charge Q and the moving device will change as a function of time. However, the distance at the time the photograph was taken can be calculated using the Lorentz transformation equations. You can use the distance formula you mentioned, but you will need to use the velocity in the x direction (βc) and the time at which the photograph was taken.