We compute the induced voltage around a 3D loop E induced by a magnetic flux change in another linked loop B.
Let di be a vector element at location x on loop E, and dj be a vector element at location y on loop B.
The magnitude of voltage induced at x in di is proportional to:
- the time rate of change of the magnetic flux in dj at y,
- the magnitude of dj,
- the magnitude of di,
- the sine of the angle between di and dj,
- 1/(distance between x and y)2
A changing magnetic flux at an element dj on B induces a voltage v at element di on E.
v = (dj×r)|r|−3 where
The magnetic field at the point which is separated from element dj by vector r is F = dj×r/|r|3.
Despite its appearance this is an inverse square law; doubling r results in quartering F.
This field F induces a voltage F · di in the element di.
We thus seek the double integral of [di dj r]/|r|3.