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I have read in my textbook that the flow of current takes place till there is a potential difference between the two points and stops when they acquire the same potential, but how can two different points have the same potential? According to me potential at a point is the work done in bringing a positive test charge from infinity to that point.

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Two different points in a circuit can have the same electric potential in the same way that two different points on a high shelf have the same gravitational potential.

  • If you lift a book up to point 1 on the shelf, you must do some work to overcome gravity.
  • If you instead lifted the book to point 2 on the same shelf, you must do the same amount of work because you lifted against the same gravity over the same distance.

You are "resisted the same amount", so to say, or you could say that it is "equally tough". So, you must apply the same amount of work. Similarly, if the "touchness" to move a charge to one point is the same as to move it to another point, then the work required to do so is the same.

The term potential energy or simply potential (a per-unit measure) is our measure for the "toughness". When it is "tougher" to reach a particular point, we say that the point is at a higher potential.

In a circuit, the "toughness" to move a charge to a point purely depends on the electric repulsion/attraction from that point:

  • It is very "tough" to place a negative charge on the negative battery terminal, for instance, because the repulsion is very large here (because a lot of negative charge is already accumulated here, and they resist further from arriving). So, we say this point is at a high potential.
  • If you attach a wire to this negative terminal, the negative charge from the terminal spreads out along the wire. When evenly spread out they will stop moving, because they mutually cancel each others repulsion out. The repulsion is therefore equally large all over this wire (if it at one point was not equal to that at a neighbor point, the charge on the wire would rearrange until it really is equal). Placing a new negative charge anywhere on this wire is therefore equally "tough" at any point. We say that the whole wire is at the same high potential at every point.

So, there is no electrical potential difference - no voltage - between two such points on this same uninterrupted wire. In the same way that there is no gravitational potential difference between two point in the same horizontal shelf. A ball placed on this shelf will not want to move to the other point; there is no tendency for it to do so. And similarly, a charge on the wire will not want to move anywhere else on this wire - no current will want to flow - since there is no tendency for it to move. Only a difference in potential gives them such tendency.

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Consider a vector field $\vec{E}:\mathbb{R}^3\to\mathbb{R}^3$. It is said to be closed if, by definition: $$\nabla\times\vec{E}=\vec{0}$$ where $\nabla\times\vec{E}$ is the curl of the vector field $\vec{E}$.

This vector field is said to be exact, if there exists a function $\psi:\mathbb{R}^3\to\mathbb{R}$ such that: $$\vec{E}=\nabla\psi$$ where $\nabla\psi$ is the gradient of the function $\psi$.

Now, in electromagnetism, we know that the electric field generated by a pointlike charge placed at the point $\vec{\xi}$ in the point $\vec{x}$ is of the form: $$\vec{E}(\vec{x})=k_0\frac{q}{||\vec{x}-\vec{\xi}||^3}(\vec{x}-\vec{\xi})$$ It can be checked that the curl of this field is zero: $$\nabla\times\vec{E}=\vec{0}$$ In general, this condition is not sufficient to conclude that there exists a potential $\psi$, since not every closed field is extact. The converse is obviously true, since $\nabla\times(\nabla\psi)=0$ for every $\psi$. In order to conclude that the field is exact, we must be in a simply connected space. In most of every physical situation, our theory is defined in a simply connected space and then we can assert that our electric field is exact, and we can define a potential.

This is the true definition of a potential. Moreover, by calculating it in different situations, it can be seen that there can be equipotential surfaces, or lines, in which every point is at the same potential.

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    $\begingroup$ Woah Kevin, I am merely in 10th standard, I didn't understand a word you said $\endgroup$ – divyam sureka Oct 17 '18 at 10:55
  • $\begingroup$ I don't see how anything before the last paragraph is needed. $\endgroup$ – Aaron Stevens Oct 17 '18 at 11:18
  • $\begingroup$ Because it leads to the correct definition of the potential.. $\endgroup$ – Kevin De Notariis Oct 17 '18 at 11:19
  • $\begingroup$ @divyamsureka sorry, didn't know that! $\endgroup$ – Kevin De Notariis Oct 17 '18 at 11:19

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