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Now, what can we do with these equations that describe electrostatic?



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It turns out that we can derive a new equation, which is called the Poisson Equation, named after



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the famous French mathematician and physicist.



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So we start from our Maxwell equation that deportation of the electric field is zero.



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So this equation is only true if we consider electrostatic because otherwise we would have here the



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time dependent magnetic fields.



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But we assume that there is no time dependence in this field.



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So the derivative is zero, which is why we have this equation here.



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So as it turns out, due to this Maxwell equation, it allows us to make an additional statement about



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the electric field e it means that we must always be able to write this electric field in terms of a



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gradient of some scalar function.



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So it means because the rotation of E is equal to zero, it must always be possible to write E as negative



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gradient of this function and this function.



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Five.



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Are we named the electrostatic potential?



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So yeah, if you want to find out what the electrostatic potential is, if you already know the electric



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field, you just have to integrate to get rid of this novel right here, which is essentially just a



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vector of partial derivatives with respect to the coordinates.



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So what you get is that your your firepower is equal to this integral here.



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And then you have here this other boundary, which could just be considered some constant.



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So as you as you know, from integration, there will always be some integration constant.



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And also it turns out that this constant doesn't really matter, physically speaking, because there's



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potential here is something like a potential energy.



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And as you know from classical mechanics, you can you can engage the zero value of your potential as



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you want.



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And it does not matter because only differences in the potential energy and also in the electrostatic



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potential matter.



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So, as I mentioned, it's quite similar to classical mechanics in classical mechanics.



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We have to force FFR and we have our potential potential energy in these two quantities are related



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by this gradient.



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And so if you want to find out what the what the potential looks like, you just have to integrate over



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the force.



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So, for example, you could have such a landscape here and you want to find out or, you know, the



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forces to to push this ball here from R zero to our then you can calculate the potential like this and



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of course, vice versa.



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If you know the potential, you can quite easily calculate the force via this equation.



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And in our case, it's very similar.



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So instead of the force we have now, the electric field and instead of the potential, we have this



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electrostatic potential.



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Now, so far, it does not really have as much to have to potential, but we also have another Maxwell



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equation, which is not concerned with the rotation of E, but with the divergence of E.



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So this is this Maxwell equation is generally true.



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It's not only true in electrostatic, but also if we have time dependent magnetic fields, for example,



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it just tells us that the sources and things of the electric fields are the charges.



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So here had divergence of E is given by the space dependent charge density ratio of R.



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And now if we take this equation and put here the electrostatic potential in this gradient instead of



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E, we end up with such an equation.



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So we get here another gradient, another Nabala operator.



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So we end up with this large class operator, which is the sum of the second partial derivatives with



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respect to the coordinates, as you hopefully remember.



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So we know now that the lower plus operator acting on this electrostatic potential is equal to this



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constant times.



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The charge density ratio of our sources means now if we have a lower charged density, Rolph, are we



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can directly calculate this this electrostatic potential and then from the electrostatic potential,



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we can calculate the electric field.



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So in some cases, this is not really helpful because it's much easier to directly calculate the electric



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field, but in other cases it is much simpler to calculate first the electrostatic potential.



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And the reason is that this electrostatic potential is a scalar function, whereas the electric field



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is a vector.



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So it has different components.



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So you already know that.



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Yeah, there's I would say less information in this electrostatic potential, which is why it's sometimes



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easier to calculate this this scalar field first.