Measurement of charge in motion

[janjan]

In Purcell's E&M Section5.3 "Measurement of charge in motion", he said when a charge is in motion, the force on test charges may not be in the direction of radius vector r. And in next paragraph, he defined Q by averaging over all directions.

However, he just measured the radial component of force and average them. Where are the non radial components?He just said the direction of force may not lie in ridius vector r. Besides why the average value of F_r is equivalent of the surface integral of E?

Also we have as yet no assurance that the force will always be in the direction of the radius vector r.
To allow for this possibility, let’s agree to define Q by averaging over all directions. Imagine a large number of infinitesimal test charges distributed evenly over a sphere (Fig. 5.4(c)). At the instant the moving charge passes the center of the sphere, the radial component of force on each test charge is measured, and the average of these force magnitudes is used to compute Q. Now this is just the operation that would be needed to determine the surface integral of the electric field over that sphere, at time t. The test charges here are all at rest, remember; the force on q per unit charge gives, by definition, the electric field at that point. This suggests that Gauss’s law, rather than Coulomb’s law, offers the natural way to define quantity of charge for a moving charged particle, or for a collection of moving charges.

 

[Vailanor]

The non-radial components of the force are not considered here since we are only interested in the total amount of charge that is passing through the sphere. The average value of Fr is equivalent to the surface integral of E because the electric field at a point is defined to be equal to the force on a test charge per unit charge. Therefore, by averaging the force on all the test charges, we obtain an average value for the radial component of the electric field, which can then be used to calculate the total charge passing through the sphere using Gauss's Law.