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the direction of electrical field in Colomb's formalism is an experimental work? can we detect the direction theoretically?

OCW Scholar - Physics II: Electricity and Magnetism
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I am absolutely sure we can. At the moment a friend of mine has a course on that. But I don't know exactly how to do it. It somehow works with the electrostatic potential. \[ \vec{E} = -\vec{\nabla}\phi\] Than the electrical field is the way a free electron would take. And the electrical field is orthogonal to equipotential. I hope that'll give you an Idea on how to solve the problem.
may be! I have to think and then continue this discussion! thanks Kathi!
it is just a definition. You just have to define it once and then stick to that definition, otherwise your theories become inconstant. After all, there is no "plus" or "minus" in nature, only two opposite charges that we needed a name for...

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Other answers:

@r0oland so you said that the base of getting direction is only theory work?
It's a theory that describes what happens in the experiment. Maybe I am not getting your question right. You define the direction of your field as going from "plus" to "minus" (ie. from the positive to the negative charge), while "plus" and "minus" are also only a definition for two opposite charges. The direction of the E-field is also the direction in which a positive test charge will feel a force. So the direction can be predicted theoretically and has also a meaning in the "real world"... I hope that helps, otherwise I would ask you to restate your question so we might find a solution... regs, Joe
hey joe! your answer educate me! but not in this case exactly! the answer of @kathi26 is exact! but I'm thankful Joe!
wikipedia quote: "The strength or magnitude of the field at a given point is defined as the force that would be exerted on a positive test charge of 1 coulomb placed at that point; the direction of the field is given by the direction of that force." Kathi26s formula just states the connection between the E-Field and the electric potential. But the electric potential is only defined using the electric field, hence we will run in circles using that. In what way is my answer not exact?
I only want a theoretical description to direction of field!

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