#### dRic2

Gold Member

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Imagine to be in 2 dimensions and you have to find the potential generated by 4 point-charges of equal charge located at the four corners of a square.

To do that I think we simply add all the contributions of each single charge:

$$V_i(x, y) = - \frac k {| \mathbf r - \mathbf r_i|}$$

$$ V(x, y) = \sum_i^4 V_i(x,y)$$

where ##\mathbf r_i## is the location of each charge. In particular if I choose the origin of the cartesian coordinates at the center of the square I get (the side of the square was set equal to 2):

##\mathbf r_1 = (-1, -1)##

##\mathbf r_2 = (-1, +1)##

##\mathbf r_3 = (+1, +1)##

##\mathbf r_4 = (+1, -1)##

Now. If I plotted it correctly I get something like this:

Which clearly has a maximum.

Now consider a circle centered at the origin but smaller than the square so that it contains no charges. Here I can write the Laplace equation:

$$\Delta V = 0$$

$$ + \text{boundary conditions}$$

A particular property of solutions of the Laplace equation is that they can have no local minimum or maximum: al extrema must occur at the boundary. This follows from the property of harmonic functions.

What am I doing wrong here ?

Thanks fro the help.

To do that I think we simply add all the contributions of each single charge:

$$V_i(x, y) = - \frac k {| \mathbf r - \mathbf r_i|}$$

$$ V(x, y) = \sum_i^4 V_i(x,y)$$

where ##\mathbf r_i## is the location of each charge. In particular if I choose the origin of the cartesian coordinates at the center of the square I get (the side of the square was set equal to 2):

##\mathbf r_1 = (-1, -1)##

##\mathbf r_2 = (-1, +1)##

##\mathbf r_3 = (+1, +1)##

##\mathbf r_4 = (+1, -1)##

Now. If I plotted it correctly I get something like this:

Which clearly has a maximum.

Now consider a circle centered at the origin but smaller than the square so that it contains no charges. Here I can write the Laplace equation:

$$\Delta V = 0$$

$$ + \text{boundary conditions}$$

A particular property of solutions of the Laplace equation is that they can have no local minimum or maximum: al extrema must occur at the boundary. This follows from the property of harmonic functions.

What am I doing wrong here ?

Thanks fro the help.

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