I The potential on the rim of a uniformly charged disk

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The discussion revolves around solving a problem from Griffiths' Electrodynamics concerning the potential on the rim of a uniformly charged disk. The original approach involved calculating the potential using a surface charge density and a specific vector formulation. A key point of confusion was the transition from the expression R^2 + r^2 - 2Rr to R^2 + r^2 - 2Rrcos(theta), which relates to the law of cosines. Clarification was provided regarding the use of the inner product in vector calculations, leading to a better understanding of the derivation. The discussion concludes with an acknowledgment of the newfound clarity in the problem-solving approach.
chaos333
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This comes from Griffiths' Electrodynamics and is problem 2.51 or 2.52, the disk has a surface charge density and my usual approach to solving these problems is to pick an area element and find a way to create a vector to the point(s) at which the potential is evaluated at. I sent a picture of my thought process and attempt at the problem. The solution involves a R^2+r^2-2Rrcos(theta) instead of R^2+r^2-2Rr that I have and I don't know how they arrived to that. Is my vector wrong or something else?
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anuttarasammyak said:
The law of cosines https://en.m.wikipedia.org/wiki/Law_of_cosines might be helpful.
I understand the answer involves the law of cosine formula, but how was that derived from the vector?
 
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Here not Rr but ##\mathbf{R}\cdot\mathbf{r}##, an inner product, for |\mathbf{R}-\mathbf{r}|^2 =(\mathbf{R}-\mathbf{r})\cdot(\mathbf{R}-\mathbf{r}).
 
anuttarasammyak said:
View attachment 347821

Here not Rr but ##\mathbf{R}\cdot\mathbf{r}##, an inner product, for |\mathbf{R}-\mathbf{r}|^2 =(\mathbf{R}-\mathbf{r})\cdot(\mathbf{R}-\mathbf{r}).
This makes a lot of sense now, thanks.
 
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