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According to Wikipedia, magnetic path length is defined as "the effective length of a closed magnetic loop inside a magnetic core made of ferromagnetic material." Thus, for a toroidal core that is wound all the way around, the magnetic path length would be \$2\pi r\$. I'm curious if this definition holds even if the core is only partially wound.

For example, if I wind a coil to drive the core 1/4 of the way around and wind another coil to measure the magnetic flux density on a non-overlapping part of the core also 1/4 of the way around, is the magnetic path length the same as the above or would I divide by 4?

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The magnetic path length is the (average) length of the magnetic field lines.

The magnetic field goes all the way around the core. The core "channels" the magnetic field since it has a higher permeability than the air. It doesn't just go through the coil, then pop outside of the core and go back to the start.

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For example, if I wind a coil to drive the core 1/4 of the way around and wind another coil to measure the magnetic flux density on a non-overlapping part of the core also 1/4 of the way around, is the magnetic path length the same as the above or would I divide by 4?

Depends on what the toroidal core is made out of, it needs to be made of iron or some other magnetic material. Magnetic materials direct the field lines, the picture below describes most of your intention and shows the magnetic field lines of a rectangular core (ignore the air gaps). This is why transformers have cores instead of air. Because of this the magnetic path length is the same for any coil on the core, and the diameter of the center would be a good approximation (although if you really get into it, you use integrals to get an exact answer)

enter image description here Source: https://physics.stackexchange.com/questions/129782/why-do-ferromagnetic-materials-channel-magnetic-field-lines

Your making a transformer, the ratio windings \$ \frac{N_p}{N_s}\$ determine the voltage (and the current) from coil to coil. The magnetic path length is the same for both coils. Since most of the magnetic field is contained in the core (there is some leakage) where you put the coils on the core is not relevant in the general sense, only the number of coils relative to one another. This assumes the core is continous,

$$\frac{V_p}{V_s}= \frac{I_s}{I_p}= \frac{N_p}{N_s}$$ Source: https://en.wikipedia.org/wiki/Transformer

enter image description here

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