2

2.2. Newton’s 2^nd Law.

The pre-relativistic notion that inertial mass (both mechanical and electromagnetic) is an invariant attribute of a particle also requires adjustment. In relativity theory,

, (2.2_1)

. (2.2_2)

The full equation of motion for a charged particle with nonzero mechanical mass then becomes

. (2.2_3)

(Here m=m_mech+m_elecmag.)

When a and v are parallel, then

, (2.2_4)

and when they are perpendicular, then

. (2.2_5)

In the case of periodic motion along the x-axis, the first "conservative" term in Eq. 2.2_3 again results in zero work per cycle:

. (2.2_6)

Only the work expended by the F_radiative part of F is nonzero.

A useful term in many calculations is "P_rad," defined to be the instantaneous rate at which F_radiative does work:

. (2.2_7)

That is,

. (2.2_9)

In the case of periodic motion,

. (2.2_10)

However, owing to time delays, one should not conclude that, at any moment,

. (2.2_11)

In the literature there is a formula for the rate at which radiant energy fluxes through an enclosing surface. It is known as the Larmor formula and has the form

. (2.2_12)

In one-dimensional cases it is customary, on the strength of this formula, to say that P is proportional to a². In other words, the assertion is that accelerating charges invariably radiate. There are difficulties with this assertion, however. For example, let us consider an oscillatory motion, x=A sin(wt). Some agent drives the oscillating particle, expending power in the amount Fv. Now according to Larmor the radiated power is maximum precisely when the acceleration is a maximum. But this happens to be when the velocity is zero!

Another objection to the Larmor formula is discussed in the next section.