Induction

Induction

Before going through the math behind induction, it is best to first look at what is going on at the atomic level. Everything is charged at that level. Electrons and nuclei have charges, they are moving, thus they have magnetic fields. Therefore everything that has matter has magnetic fields. Magnetic fields are (so far as we know) infinitesimal in size so they are affected by all matter. The size of the effect is additive, and since magnetic fields have direction, these sums can be large or small depending on the directions of the fields for each particle. Since magnetic fields interact with each other, everything is interacting with everything else. Actually there are 4 fundamental 'interactions'. These are called gravity, electromagnetism, the strong force and the weak force. These forces work over their own ranges, becoming infinitesimal outside those ranges.

At the sizes and distances of atoms and free electrons, the dominant force is electromagnetic. When an external magnetic field moves through matter, the field is affected by the matter and the matter is affected by the field. This is the fundamental cause of induction. Sample A is affected by Sample B via their magnetic fields. A basic example of action at a distance.

For samples with randomized magnetic fields, the sum of all the fields is close to zero, so this sample will have small effects on other samples via magnetic fields since their net magnetic fields are small. For samples with large net magnetic fields, or with potentially large magnetic fields (i.e. the sample can be manipulated in some fashion to generate a field, such as passing current through a wire), there are large effects via magnetic field interactions.

Extrapolating this idea, a current passing through a wire of sample A will generate a magnetic field, which will interact with a wire of sample B that is close by and moving with respect to sample A, generating a magnetic field which will cause a current to flow in the wire of sample B. This is induction. The current in sample A induces a current in sample B, and it happens via magnetic fields. They key part of this experiment is that the magnetic field from sample A needs to be moving relative to the wire in sample B. It is the variability in the magnetic field, not just the field, that causes induction.

Math of Induction

If induction is looked at using the method described above, the math should be simple: Use an equation that gives the strength of a magnetic field over space generated by a particular current in sample A, then use the inverse of this equation to get the current induced by a magnetic field at a certain location in sample B. Complications occur because everything varies with time and spatial location, but these can be handled with integrals and partial derivatives.

Induction In NMR

NMR uses induction by having a small antenna close to the sample that has current induced in it when the net magnetic fields in a sample relax after excitation. In this case you have moving magnetic fields and stationary conductor.