RETARDED POTENTIALS CONTINUED

RETARDED POTENTIALS CONTINUED IF ρ IS CHANGING WIH TIME, CONSIDER THE FIGURE BELOW P HAVE POSITION VECTOR r AND Q HAS POSITION VECTOR r’ IF CHARGE DISTRIBUTION NEAR Q CHANGES WITH TIME, THE INFORMATION THAT IT HAS CHANGED CAN ONLY BE APPRECIATED AT POINT P AT TIME |r-r’|/C AFTER THE CHANGE. THIS IS THE TIME IT TAKES THE ELECTROMAGNETIC RADIATION TO TRAVEL THE DISTANCE |r –r’| HENCE IT IS PLAUSIBLE THAT AT A TIME t THE CONTRIBUTION TO THE POTENTIAL Φ(r,t) AT P FROM THE CHARGE WITHIN THE VOLUME ELEMENT 𝜕τ’ AT Q DEPENDS NOT ON WHAT THE CHARGE IS AT TIME t BUT ON WHAT IT WAS AT TIME t-|r-r’|/C THIS IS TRUE FOR ALL THE VOLUME ELEMENTS WITHIN THE VOLUME V. NOTE THAT C IS THE SPEED OF LIGHT WHICH IS ALSO THE SPEED OF PROPAGATION OF ELECTROMAGNETIC WAVES AND FIELDS. NB: THE TIME LAPSE |r-r’|/C DEPENDS ON THE POSITION VECTOR r’ AND HENCE VARIES OVER THE VOLUME V Φ(r,t) = (1/4πε0)∫v((ρ(r’,(t-|r-r’|)/C)/(|r-r’|)).dτ’) (&&&&&) Similarly THE SOLUTION TO THE EQUATION(&&&) GIVES POTENTIAL A THIS IS GIVEN BY A(r,t) = (𝝻0/4π) Φ(r,t)∫v((j(r’,(t-(|r-r’|)/C)/(|r-r’|)).dτ’) (&&&&&&) THE SOLUTION TO EQUATION (&&) HENCE OBTAINED BY INTEGRATING THE CONTRIBUTIONS OVER THE VOLUME V