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The test charge is used to measure the electric field. The problem is that if we measured some value of the field, then the test charge is removed, what will happen? The measured field is still there? Or it will be changed or it will completely disappear?

I have heard there are a theory called "action-at-a-distance" and "absorber theory", they said when the test charge is removed, the measured field also gone! Hence, we can never measure a correct electric field? Is this true?

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We talk of fields as though they are there even when we've not got a test charge in place to detect their presence.

If we clung to an action at a distance mechanism, when we said there's a field at point P we'd mean that IF we put a test charge at P it would experience a force. [This would be by direct action of the source; the only role the intervening space would play is to distance the source from the test charge.]

If we believe in fields as physical things (and, these days, who doesn't?) then we believe that something is different about the space containing point P compared with when there's no field, even when there isn't a test charge at P. If we did put a test charge at P it would, of course, experience a force. [This time we attribute the force to the altered condition of the region that we've put the test charge in, brought about by the source.]

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For historical and didatical reasons we talk about "test charge". Today we think in terms of fields, but in the early days of eletromagnetism they thought in terms of forces. For me the test charge is not different of a hypothetical observer in mechanics.

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