Reference Frame

98

General Theory of the Electromagnetic Field

in the reference frame K o if the point-like charge is at rest in this reference frame and:

1F q 1E , (2.2 b)

in the reference frame K1 if the point-like charge is at rest in this reference frame.

The vector quantities Eo and E1 represent the electric field strengths in the reference frames K o and K1 , respectively.

If the point-like charge (or the body represented by this charge) is moving relatively to the reference frame, then the law of ponderomotive action is different from that presented above. In this case, the law can be established either by the generalization of experimental results or by using certain expressions from Mechanics established in the Special Theory of Relativity.

2.2. DERIVATION OF THE EXPRESSION OF THE LAW OF PONDEROMOTIVE ACTION UPON A POINT-LIKE ELECTRIC CHARGE THAT IS MOVING RELATIVELY TO AN INERTIAL REFERENCE FRAME

The expression of the law of ponderomotive action upon a point-like charge that is moving with respect to an inertial reference frame can be established by using certain assumptions (hypotheses) from Mechanics established in the Special Theory of Relativity.

The used hypotheses are the following: 1. In an inertial reference frame, the expression of the force acting upon a point-like electric charge that is in motion relatively to this reference frame, and produced by the interaction with another point-like electric charge, at rest with respect to the same reference frame, is given by the Coulomb law from Electrostatics. 2. For obtaining the expression of the force acting upon a point-like electric charge the principle of superposition will be used. So, in order to find the force acting upon a point-like electric charge q at rest or in motion, under the action of several point-like charges qi ni .,..,1 , we shall proceed as follows. We consider separately the pairs of charges q, 1q ; q, q2 ; . . .; q nq , and we obtain the forces 1F , F2 , . . ., Fn . The resultant force exerted upon the point-like electric charge q is:

.F

n

i 1 iF (2.3)

The subscript of point-like electric charges indicates only the ordinal number and has no relation with the subscript of the reference frame symbol. On the other hand, the definition of the electric charge does not depend on the reference frame. 3. For expressing the forces in another inertial reference frame, the transformation relation of forces, from Mechanics, established in the Special Theory of Relativity, will be used.