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Re: Potential of a point charge



I think this confusion is widespread, but I find that it pretty much
evaporates if one keeps in mind two things:

1) The integral involves a dot product of the VECTOR E_vec with the
VECTOR ds_vec.

2) When integrating "radially" the vector ds simply becomes the
vector (dr r_hat) where r_hat is a unit vector in the positive r
direction (i.e., radially outward) and dr is a SCALAR with an
algebraic sign that is determined AUTOMATICALLY by the limits of the
integral. If the lower limit is larger than the upper limit, than dr
IS negative. One need not "make adjustments" for this fact by
putting in a minus sign somewhere; its all "in there" already.


Thus, to find the absolute potential at a distance R from a point
charge, we simply apply the definition

delta V = V_B - V_A = - int( E_vec dot ds_vec from A to B)

to a radial path from "infinity" (where V = V_A = 0) to R where (V = V_B)

V = - int( [k q r_hat/r^2] dot [dr r_hat] from infinity to R)

= - k q int( [dr/r^2] [r_hat dot r_hat] from infinity to R)

= - k q int( dr/r^2 from infinity to R)

= - k q (-1/r evaluated from infinity to R)

= k q/r

--
John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm