inverse square law for the force exerted by a stationary charge[1].
Consider a set of coordinates S(x, y, z, t), and let there be a charge
in uniform motion along the X axis with the velocity
. We desire to know the force acting at the time
on any other charge
, which has any desired coordinates
and
and any desired velocity
.
Assume a system of coordinates
moving with the same velocity as the charge
which is situated at the origin. To an observer moving with the system S', the charge always appears at rest an to be surrounded by a pure electrostatic field. Hence in system S' the force with which
acts on
, will be in accord with Coulomb’s law.
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or
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(18)
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(19)
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(20)
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where
and
are the coordinates of charge
, at the time
. For simplicity let us consider the force at the time
, then from transformation equations (1)-(3) we shall have
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Substituting into (18), (19), and (20) and also making use of the transformation equations of force (15), (16), and (17), we obtain the following equations for the force acting on
, as it appears to an observer in system S.
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(21)
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(22)
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(23)
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- ↑ In its simplest form, Coulomb’s law merely states the force acting between two stationary charges. It should be noted that our derivation assumes the same law for the force with which a stationary charge acts on a moving charge.