Electrical potential and potential difference

[ajay.05]

Indian school books say -- We define the electrical p.d. between two points in an electrical circuit carrying some current as the work done to move a unit charge from one point to other!
=>V=W/Q

But, my doubt is...since work done is directly proportional to distance between two points, shouldn't the p.d. change from point to point(depending on their distances)?

 

[voko]

Generally, the potential difference depends on the geometry of the problem, which includes distances between points. But in a large number of problems, which includes steady-state DC circuits, the p.d. is independent of that, or the dependence is negligible.

Take, for example, a large sphere that is connected to a pole of a battery. Each point of the sphere will have a p.d. with the other pole of the battery. But what is the p.d. between any two poles on the sphere?

 

[Delta2]

For a conservative electric field (and for all practical purposes the electric field in circuit theory is considered to be conservative) the work of the electric field between two points depends only on which are those two points and not in the trajectory we choose between those two points.

That is because since the field is conservative then E=\nabla \phi(r) and since the work between two points is defined as the integral W=\int_{\gamma(p,q)}{E(r)dr} (where \gamma(p,q) is any curve between the two points p and q) which integral W by the gradient theorem http://en.wikipedia.org/wiki/Gradient_theorem depends only on the value of the function \phi(r) at the points p and q.