If you want to know the value of the electromagnetic field at some point in space P at time t1, I assume that since EM is a relativistic theory, it should be possible to derive it using only the value of the field (along with charges, but let's say we are dealing with fields in free space) at an earlier time t0, in a sphere of radius c*(t1 - t0), with the field outside the sphere unknown. Does anyone know of any books or papers that show what the equations for this would look like, and how they would be derived, preferably at a not-too-advanced level? In my EM class we did study the "retarded potential" method which is based on the past light cone, but it requires looking at the contribution from every charge whose path crosses the past light cone at any point in the arbitrarily distant past, whereas I'm thinking of a method that only looks at a cross-section of the light cone at a particular earlier time t0, and which deals with the field at that time rather than looking exclusively at the sources of the field (charges).
Another question related to this: if you know the initial conditions I in the sphere at time t0 (both the E and B fields, and I think you'd need their instantaneous time-derivatives to determine their later evolution too), I wonder if you could use the following approximation to calculate the time-evolution, an approximation which would hopefully approach the exact solution in the limit considered in step 4:
1. Divide the whole volume of the sphere up into a lattice of cubes (or any other space-filling volume), and find the average value of the fields in I within each cube.
2. Create a new set of initial conditions I' where the fields have uniform values within each individual cube (with discontinuities at the boundaries between cubes), equal to the averages found in step 1.
3. Compute the time-evolution of [i]I'[/b] using the superposition principle--the total value of the field at a later time is just the sum of the contributions from a large number of separate initial conditions, each one consists of a single cube filled with uniform fields surrounded by an initial field of 0 everywhere outside that cube.
4. Then, consider the limit as the size of the cubes goes to zero. I would guess that in this limit, the contribution from each cube might approach some fairly simple form, perhaps identical to the contribution from a proportionally small spherical region of uniform field with an initial field of 0 everywhere outside the small spherical region. Probably the time-evolution of such an initial condition would just be a sphere of nonzero field expanding at the speed of light, though I don't know whether the field would be uniform in the expanding sphere or concentrated at the surface.
Could this work as a way of deriving the time-evolution, or is there some problem I haven't considered?
Another question related to this: if you know the initial conditions I in the sphere at time t0 (both the E and B fields, and I think you'd need their instantaneous time-derivatives to determine their later evolution too), I wonder if you could use the following approximation to calculate the time-evolution, an approximation which would hopefully approach the exact solution in the limit considered in step 4:
1. Divide the whole volume of the sphere up into a lattice of cubes (or any other space-filling volume), and find the average value of the fields in I within each cube.
2. Create a new set of initial conditions I' where the fields have uniform values within each individual cube (with discontinuities at the boundaries between cubes), equal to the averages found in step 1.
3. Compute the time-evolution of [i]I'[/b] using the superposition principle--the total value of the field at a later time is just the sum of the contributions from a large number of separate initial conditions, each one consists of a single cube filled with uniform fields surrounded by an initial field of 0 everywhere outside that cube.
4. Then, consider the limit as the size of the cubes goes to zero. I would guess that in this limit, the contribution from each cube might approach some fairly simple form, perhaps identical to the contribution from a proportionally small spherical region of uniform field with an initial field of 0 everywhere outside the small spherical region. Probably the time-evolution of such an initial condition would just be a sphere of nonzero field expanding at the speed of light, though I don't know whether the field would be uniform in the expanding sphere or concentrated at the surface.
Could this work as a way of deriving the time-evolution, or is there some problem I haven't considered?
