Section 3.3 · Home exercises · 6 of 9
Home exercise 3.3.6: Work along two paths
Work along two paths
#Find the work done in carrying a charge from to in the field
(a) along the parabola ,
(b) along the straight line joining and .
Compare the two results.
Compute directly on each path: express , and through a single parameter ( on both paths). If the field is electrostatic, both answers must agree.
The work done against the field is, by Eq. (3.3.11),
(both paths lie in the plane , so ).
On we have , so
With running from 1 to 2,
The line through and is
so that
With running from 1 to 2,
Units: . The two paths give the same work, and the potential gives it without any line integral (remark below).
The result is independent of the path, as it must be for an electrostatic field: is curl-free () and equals with . Then directly, without any line integral. The -coordinate plays no role because has no -component. Positive work is needed because the negative charge is carried to a point of lower potential.
Deform the path in any way you like between the same two end points: the line integral of a curl-free field does not change.