Searching for Laws of Physics: The Electromagnetic Field Tensor
How can we be sure that the laws of electrodynamics (Maxwell's Equations) are consistent with the Special Theory of Relativity?
QUESTIONS
(Score is number right minus number wrong.)
An example of a Lorentz invariant or proper quantity is
Equation (3a).
Equation (3b).
Equation (3c).
Equation (3d).
An example of a Lorentz 4-vector is
Equation (3a).
Equation (3b).
Equation (3c).
Equation (3d).
All of the quantities defined in Equation (4) are 4-vectors. They all inherit their transformation properties from Equation (1) since they have been created only by multiplying or dividing Equation (1) by an invariant or proper quantity. They are all almost the simplest things you can imagine creating from the original 4-vector. They all become elements from which to build potential laws of physics by incorporating them into 4-vector or 4-tensor equations.
If we had been astute enough in Maxwell's time to understand Special Relativity and the idea that sources (charges and currents) can generate fields, we might have made the mental jump to define a 4-potential from the relationship of Equation (5), i.e., given the 4-current from Equation (4) already derived and the principle that laws of physics must be expressed as 4-vector relationships. Viewed this way, the laws of electrodynamics are about as simple as one could imagine!
Which of the following has exactly six independent components?
4-vector
Two 4-vectors
4-Tensor
An antisymmetric 4-tensor
A symmetric 4-tensor
Two 4-Tensors
How many independent components does a symmetric 4-tensor have?
3
4
6
10
16
How many independent components do the electric and magnetic fields have, taken together?
3
6
10
Which defines an antisymmetric 4-tensor?
Equation (6a)
Equation (6b)
Equation (6c)
Of the 4-vector equations that might be candidates for 'laws of physics', which is a 'gauge condition'?
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Of the 4-vector equations that might be candidates for 'laws of physics', which is a 'dynamic law of motion' (Newton's Second Law)?
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Of the 4-vector equations that might be candidates for 'laws of physics', which is equivalent to the definition of the 'Lorentz Force'?
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Of the 4-vector equations that might be candidates for 'laws of physics', which contains Equation (13)?
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Of the 4-vector equations that might be candidates for 'laws of physics', which contains Equation (14)?
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Of the 4-vector equations that might be candidates for 'laws of physics', which expresses the 'conservation of charge'?
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
Of the 4-vector equations that might be candidates for 'laws of physics', which is equivalent to Equation (11) and Equation (12)?
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
Equation (7e)
Equation (7f)
Equation (7g)
Equation (7h)
The placement of the electric and magnetic field components into the electromagnetic field tensor (see Equation (15)) is not arbitrary. They must be placed in such a way that the Lorentz Force (Equation (7c)) used in the dynamic force law (Equation (7g)) is consistent with experiment. Ultimately, the way we measure fields is to place a test particle in the field and observe its motion. Where we place the things that we have called electric and magnetic field into the field tensor is determined by this kind of experiment. They must be placed so that they account for the motion of a test particle through Equation (7g). The electric field and the magnetic field are then best thought of as components of the same thing, the electromagnetic tensor field.
EQUATIONS
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