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I've solved many problems calculating the electric field, $E$, of a given charge distribution but I'm just musing a bit as to why this is useful. The definition of $E$ involves the force, $F$, on a test charge, $q$, and the ratio $F/q$ in the limit as $q-> 0$. So if we have an $E$ field and place a non-zero charge in it, unless the source distribution is glued in place and/or the charge is very far away the charge will cause redistribution of the source charges and the resultant force on the introduced charge will not be $qE$ using the $E$ originally calculated.

So why is the concept of the electric field so useful? Of course, we could dispense with the $E$ field and directly compute the Coulomb force on a non-zero charge due to a given configuration of other charges, but how realistic are all these problems if the effect of the "test" charge on the source configuration is neglected?

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One of the reasons is the problems with the concept of action at a distance. We could say that moving charges in spacecraft at the asteroid Bennu caused forces that move charges at the NASA control in earth. And call it eletronic communication.

But the delay of that action suggests that something (EM waves composed by E and B fields) travelled between the 2 locations.

Coulomb and Ampére followed the Newton approach of action at a distance, but the discovery of EM waves changed the game in favour of field approach in my opinion.

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So if we have an E field and place a non-zero charge in it, unless the source distribution is glued in place and/or the charge is very far away the charge will cause redistribution of the source charges and the resultant force on the introduced charge will not be qE using the E originally calculated.

Not true.

Another way to keep to keep the electric field constant without fixing the charge in place or moving it far away is to use conductive materials to hold the charge, and arrange to maintain a constant potential difference between them. In this scenario the charge is free to move around, but as long as the test charge (the charge acted on by the field) is small compared to the charge establishing the field, the field between the charged regions (which we can call electrodes) will be essentially constant.

This is directly relevant, for example, to the operation of vacuum tubes.

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  • $\begingroup$ But you're including the condition that the charge acted on by the field is small compared to the charge establishing the field. $\endgroup$ – Not_Einstein Oct 25 '20 at 1:04
  • $\begingroup$ @Not_Einstein this condition applies to many useful situations, therefore requiring it does not make the concept of electric field useless. $\endgroup$ – The Photon Oct 25 '20 at 2:48

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