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Potential difference between two points is defined as the work done in moving a net positive charge from one point to the other. But, my question is, when the two points, that are potentially different, are connected using conducting wire having length more than the distance between the two points, charge flows through the conductor; and since the charge cannot move directly through air to the other point, it has to travel all the way through the length of the conductor, and that means it has to move a longer distance than the actual distance between the two points. Therefore, more amount of work would be required in order move that charge from one point to the other since the charge has to travel a longer distance. Thus, it would mean that the potential difference between the two points increases with the length of conductor.

Work done, as we all know depends on displacement not distance, but moving anything through a longer distance would require a greater amount of work. Where am I wrong?

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  • $\begingroup$ Work done by a constant force does depend on distance. But, if the wire is longer is the force the same? Consider that question. $\endgroup$ – Jon Custer Jan 4 at 2:32
  • $\begingroup$ Suppose, I pushed a wooden block from one point to the other not through the direct path but by following a rectangular path, then I would do a lot more work on block than I would have done if I had pushed it through the direct path. Does not it mean that work done depends on the path taken? $\endgroup$ – Suyash Ishan Jan 5 at 13:44
  • $\begingroup$ Yet that is not the situation in this case. The longer the path, the less the force in the situation you discuss. Ultimately it is probably easiest to look at it from an energy rather than force perspective. $\endgroup$ – Jon Custer Jan 5 at 14:58
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When we refer to work done in such a situation, we actually refer to the change in potential energy of the unit charge used. Using a wire may complicate things a bit. So let's just assume we have to move the test charge between two equipotentials without giving that charge any kinetic energy in the process. So we'll only consider the work we did to change the potential energy of that charge.

enter image description here

Imagine moving the charge from equipotential $A$ to $B$ across the two different field lines $1$ and $2$.

We can clearly see that though in path $1$ we have to push the charge across a greater distance than in path $2$, we have to apply less force at each point during our journey than in path $2$.

$$W_\text{us}=\int\mathbf F\cdot d\mathbf r=\int Fdr\cos0=\int Fdr$$ (because we are moving across a field line)

So these two factors balance and hence we have to do the same amount of work to get the charge from contour $A$ to contour $B$.

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Work W is a scalar quantity defined as the scalar product of two vectors, Force F and distance or displacement x. In other words, dW = F.dx. The unit positive charge of your example can travel to kilometers without doing any work if it moves perpendicular to the force field. For voltage points V1 and V2, the work done against field E is: W = -Integral E.dx as carried out from point 1 to point 2 of any short or long path and will always be equal to V2-V1 independent of the path since the electrostatic field is conservative field of force

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