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So in this section, we want to calculate for the first time the electrostatic potential, and there



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we go back to an example that we have already solved, for example, a sphere that is charged.



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So we have some rotational symmetry in our charged distribution.



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So in the previous section, we have derived this song equation where we have used this Maxwell equation



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that is only true in electrostatic.



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And it allowed us to introduce such an electrostatic potential.



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And with the help of the other Maxwell equation for the electric field, we ended up with this equation



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here, which is second order in the coordinate because this is yodeler plus operator.



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Now, when one of the previous lectures we have already derived to Cullum's law for the electric field.



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So what we had considered here is, for example, a charged sphere which is charged by the charge.



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Q Or we could also labeled.



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Q If we if we as we have done here and we what we basically did is we have considered this integral



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formulation of the Maxwell equation, where this is the left side and this is the right side and the



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right side just simplifies to the inclosed charge in this volume.



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B now since we have this rotational symmetry, we know that our electric field is always pointing along



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the radial direction, which is why we couldn't write it down as such a scalar function times this vector



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of art.



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And so this means that this E is always parallel to this integration, uh, vector here D, which is



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perpendicular to the surface of our I've always feary that we consider.



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And so it's very easy to calculate this integral.



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This is just the surface of our sphere.



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So it's just four pi r square.



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And so straightforwardly we got the electric field which is proportional to one of our square.



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And then here in this vector formulation, we just have to take this position back to each of our into



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account.



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So this was how we calculated the electric field.



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Now we if we have the electric field, we can also calculate the electrostatic potential by using not



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this equation because yeah, if we would solve this puzzle equation, we would have to start again from



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zero.



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So instead, if we already have Yovani and it's very easy, we just use this equation.



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So we just integrate over E and from from some.



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Yeah, from some point of reference are zero, which is not really important.



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But we integrate up to the position vector R.



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So hopefully you remember some integration laws, so if you have one over a square dependent's and you



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integrate it, you get one of our with a minus sign and then there is another minus sign here, which



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is why we end up for the electrostatic potential with one over four Pi Epsilon zero times to charge



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over R.



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And then of course we get this other term, which you could consider as an integration constant, for



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example.



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And if you remember this analogy to the classical world, we have to remind ourselves that only differences



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matter in this quantity because it is something like a potential energy.



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So what we can do is just we can take this integration constant and set it to zero.



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Or we could also say we shift our whole electrostatic potential by some term so that this term here



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cancels out.



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And so this is what what what it makes sense to consider the electrostatic potential in this case.



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But you could also write down that the electrostatic potential is dysfunction, plus an impossible constant.



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So now we know the functional behavior of this electrostatic potential and it looks quite similar to



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to the electric field.



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But instead of a one of a R-squared dependence, we have here at one of our dependence.