Question

show that work done in an electric field is independent of path

Answer 1

A point charge moves from point to point in the field of an opposite electric charge. You can modify the path taken by dragging the locators. At each point of the path, the charge experiences a force that can be represented by a vector (orange) that makes an angle with the differential displacement (blue). Summing the dot product of vectors along the curve (summarized on the right side) gives the total work performed by the charge. The total work does not depend on the shape of the curve, only on the initial and final points. In this case, work is done only by the radial displacement, the dot product with the tangential component being zero. Work is independent of path only for a conservative field, which includes electrostatic and gravitational fields.