Electrostatic Problem at point on a conical surface

[roam]

Homework Statement

I am trying to understand a solved problem which is about finding electrostatic potential at point b of the following conical surface with a given surface charge:

I have attached the worked solutions to this post. In the solutions, I don't understand how they have got the expression:

$\bar{r}=\sqrt{h^2+r^2-\sqrt{2}hr}$

The Attempt at a Solution

I appreciate it if anyone could explain how this expression was obtained.

Looking at the expression it looks like Pythagoras was used here with $\bar{r}$ being the hypotenuse. But when the vertical side is $h$, how do we get $r^2-\sqrt{2}hr$ as the other side? I'm very confused here.

Any helps is greatly appreciated.

 

[TSny]

The triangle formed by $h$, $r$, and $\bar{r}$ is not a right triangle. Can you see how the law of cosines can be used to get the expression for $\bar{r}$?

 

[roam]

The triangle formed by $h$, $r$, and $\bar{r}$ is not a right triangle. Can you see how the law of cosines can be used to get the expression for $\bar{r}$?

No, how can I use the law of cosines when the triangle is not a right angle? Should I be considering a different triangle? I really have no idea how the expression for $\bar{r}$ was obtained.

 

[BvU]

express r in $\mathcal r$

 

[TSny]

No, how can I use the law of cosines when the triangle is not a right angle? Should I be considering a different triangle? I really have no idea how the expression for $\bar{r}$ was obtained.

The law of cosines applies to all triangles, not just right triangles.

See http://www.mathsisfun.com/algebra/trig-cosine-law.html

 

[roam]

Thank you so much BvU for the diagram. It makes perfect sense now.

Thank you Tsny, I really appreciate the link.