In place of Coulomb's Law,F=q1q2/4φε0r2,the formulaF= (q1q2/4φε0r2)√[1+v2/c2]is proposed, wherevis the velocity of one charge relative to the other. Using this velocity-dependent force law, it is shown how Ampere's law can be derived; Maxwell's equations can then be obtained, and their limitations can be discussed. The concept of field is shown to be valid only in the limiting case of vanishing velocities, so that Maxwell's theory is approximately valid only when any motion is slow and any rate of change of field is small. When these conditions do not hold, Maxwell's theory is invalid, but the velocity-dependent force law can be used. The motions of charged particles in electric and magnetic fields are discussed. Anomalous apparent decay times of fast mesons are attributed not to time dilation but to speeds greatly exceedingc. Some anomaly remains, and it is suggested that new experimental observations are necessary to resolve this.