How did Tesla measure electromagnetic fields, etc?

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Macoleco
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He didn’t have the modern equipment we have today.
 
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You can be reasonably precise with variations on the compass needle.
A lot of work was already done by C.F. Gauss and published in 1832.
https://www.jstor.org/stable/pdf/110344.pdf
 
It depends a bit on the time period since Tesla was active when many of these instruments were developed. Quite a instruments can actually be built using electromechanics (Moving coil multiimeters are still used) and once you introduce vacuum tubes (early 20th century) you can build simple versions of many "modern" electronic instruments.
By WWI you could e.g. get pretty good wavemeters etc
 
Hertz did it by measuring the length of spark at the receiver, which was a resonant loop with a spark gap.