18. Separating the transformed Maxwell equations
In Fig. 18.1 below, we restate the transformed Maxwell Equations as derived in the previous two chapters:
The first Maxwell equation as transformed: Chapter 16, Fig. 16.5 | |
The second Maxwell equation (a vector equation) as transformed: Chapter 17 | |
The third Maxwell equation as transformed: Chapter 17 | |
= 0 | The fourth Maxwell equation (a vector equation) as transformed: last equation in Chapter 17 |
Fig. 18.1. Transformed Maxwell Equations as derived in the previous two chapters. |
Unfortunately Maxwell's equations transformed as mixes of Maxwell's equations in the prime reference frame. We now need to separate the mixes. In order make the equations more manageable, we will use the abbreviations shown below in Fig. 18.1a. Note that each of the newly defined quantities are the left sides of the wished for Maxwell equations in the primed frame. We hope to show that the above equations (in Fig. 18.1 above) imply that all these quantities listed in Fig. 18.1a are zero.
Using the definitions in Fig. 18.1a, we can be expressed the equations in Fig. 18.1 as:
We need to solve the above equations for the variables m1, m2_{x}, m2_{y}, etc:
Inspecting the equations, we see that four of the variables are equal to zero, i.e. (C2y), (C2z), (C4y), and (C4z). Thus ⇒ | m2_{y} = m2_{z} = m4_{y} = m4_{z} = 0 |
Multiplying (C1) by V and subtracting it from (C4x) yields
| m4_{x} = 0 . |
Substituting m4_{x} = 0 into (C1) yields ⇒ | m1 = 0 . |
Similarly, we can multiply (C2x) by V and add this to (C3) to yield: | m3 = 0. |
Inserting m3 = 0 into (C2x) gives ⇒ | m2_{x} = 0 . |
Fig. 18.3. Solving for m1, m2_{x}, m2_{y}, etc. |
In summary, all eight of the the variables m1, m2_{x}, m2_{y}, ... , m4_{z} are zero. Using the definitions of these variables as listed above in Fig. 18.1a gives:
Thus, we have successfully shown that the Lorentz transforms when operating on Maxwell's equations, produce the same equations in the new reference frame.
Most text books would do this derivation much more efficiently using more advanced methods, briefly mentioned in Chapter 14. We have presented here this vector algebra derivation as an alternative, in part to illustrate that most physics can be derived in a number of ways and we can learn from each way.