Q.Explain, using classical electromagnetic theory, why Rutherford's model predicts that an orbiting electron should spiral into the nucleus in a fraction of a second, and explain why this prediction is a serious problem for the model.

Classical electromagnetic theory states that any ACCELERATING charge radiates electromagnetic energy continuously. An electron moving in a circular orbit is always accelerating, since its VELOCITY DIRECTION is constantly changing even if its speed is constant (centripetal acceleration $v^2/r$). Classically, then, the orbiting electron must be continuously emitting radiation and therefore continuously LOSING energy. Since the electron's orbital energy depends on its orbital radius (Section 1.5's $E\propto-1/r$), a continuous loss of energy forces the radius to continuously SHRINK -- the electron does not stay in one orbit but spirals slowly, then increasingly rapidly, inward. Working this spiral out using classical mechanics and classical radiation formulas gives a total collapse time of only about $10^{-8}\ \text{s}$ for the electron to reach the nucleus.

This is a fatal problem for a purely classical Rutherford atom: real atoms are observed to be stable, persisting essentially unchanged for the lifetime of the universe, not collapsing in a hundred-millionth of a second. Bohr's first postulate resolves this directly by simply DECLARING that electrons in certain special (stationary) orbits do not radiate, sidestepping the classical prediction rather than deriving its failure from a deeper principle.

State the classical rule (accelerating charge radiates), apply it to an orbiting electron, follow the consequence (shrinking orbit, quoted collapse timescale), then contrast with the observed fact.

• Thinking "accelerating" means the electron's SPEED must be changing -- it is the changing DIRECTION (centripetal acceleration) that matters for a circular orbit.
• Omitting the actual (approximate) collapse timescale, which is the concrete, memorable number examiners look for.