# How can I calculate in MATLAB-the electric field due to a time varying magentic field for a simple geometry?

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Md Chowdhury on 11 Aug 2021
Edited: David Goodmanson on 12 Aug 2021
I have a problem of electromagentic modeling. The problem involves a current carrying circular coil where the current will be varied with time. Calculating the magnetic field was relatively simpler - I used Bio-Savart's law & the superposition theory. But I am facing some difficulty calculating the electric field that arises from the time varying magnetic field, specifically, I didn't know how to implement Maxwell's equation numerically. I would appreciate any help in any form, however, after much speculation, I finished writing my first script of calculating the electric field. I amm attaching the vector plot of the electromagnetic field distribution in a 3D volume. I am also attaching the .fig file of the corresponding plot so anyone interested can maneuver plot as they like. I would like to know if the electric field I calculated is correct.
G A on 11 Aug 2021
https://www.mathworks.com/matlabcentral/fileexchange/54557-3d-finite-different-time-domain-first-order-mur-boundary-condition

David Goodmanson on 12 Aug 2021
Edited: David Goodmanson on 12 Aug 2021
Hello MC,
From the plot it is not that easy to see what is going on since the two field colors are similar. Could you post a separate plot for each field?
If you can successfully produce the B field by using a Biot-Savart integration then you can produce an E field in the same way, especially since it's easier. Here I use the common assumptions  the current is changing slowly, so that there is a negligible amount of change to the system in the time it takes for light to propagate across the loop. Therefore retarded time effects can be neglected,  the current has the same value all the way around the loop so that there is no resulting charge density that can be the source of an electric field. In that case the B field is
B = (mu0/4pi) Integral J(r',t) x (r-r')/|r-r'|^3 d^3r'
where r,r' and J are vector quantities and (r-r') extends from the source point to the observation point. If the current is confined to a small filament with constant current I, then
B = (mu0/4pi) I Integral ds' x (r-r')/|r-r'|^3
where vector ds' is a line element that is everywhere tangent to the loop. I assume you are doing an integral like this to find B.
The electric field in this circumstance is
E = -(mu0/4pi) Integral 1/|r-r'| dJ(r',t)/dt d^3r'
and
E = -(mu0/4pi) dI/dt Integral ds'/|r-r'|
For an oscillation at a single frequency, described by e^(-iwt), then d/dt can be replaced by -iw in which case
E = (mu0/4pi) iw Int 1/|r-r'| J(r') d^3r'
and
E = (mu0/4pi) iwI Int ds'/|r-r'|
The factor of i just means that B and E are 90 degrees out of phase as functions of time.
The E field lines go in circles and look similar to the current loop.

R2016a

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