Confused about the boundary conditions on a conductor

[bubblewrap]

In the textbook (attached image) it says that the boundary condition is V=0 at r=R.
This creates a correlation that

$B_l=-A_l R^{2l+1}$

but the potential at any boundary is continuous so when we take this account, we get.

$B_l=A_l R^{2l+1}$

These two clearly contradict each other. I'd like to know why this contradiction occurred.

Many thanks.

 

[BvU]

Does the textbook also tell us what A and B are ?

I can't really see, the picture is so vague
https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686781/

but the potential at any boundary is continuous so when we take this account, we get.

$B_l=A_l R^{2l+1}$

How so ? can you elaborate ?

 

[Charles Link]

See http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinates.html
I think you are ignoring the possibility of an $l=0$ solution. $\\$ You then have a separate solution inside the sphere that says $V=0$, that has both $A$ and $B =0$. $\\$ The coefficients of $A$ and $B$ inside the sphere are different from what they are outside. The potential $V$ is continuous across the boundary, but that doesn't mean that each of $A$ and $B$ stay the same. $\\$ Your first equation is correct, but for this case ($r \geq R$) you only have the $l=0$ term. I'm not sure how you came up with your second equation. $\\$ You really need to label the coefficients $A_{l \, out}$, $B_{l \, out}$, $A_{l \, in}$, and $B_{l \, in}$. $\\$ Edit: I looked at it a second time=I misread the problem. I thought $V=0$ everywhere at $r=R$ . Let me try again... It would help if you could show us the previous page, because otherwise it is a guessing game of what we are trying to solve.

 

[bubblewrap]

Does the textbook also tell us what A and B are ?

I can't really see, the picture is so vague
https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.686781/

How so ? can you elaborate ?

Since the term $r^{l}$ diverges as r>>R, it is not considered at r>R, similar logic was also applied for $r^{-l-1}$ for the case inside the sphere.

Since both terms should have the same value at r=R (since the potential is continuous at any boundary) the terms $A_l R^{l}$ and $B_l R^{-l-1}$ should have the same value, hence the correlation mentioned above.

However, if we consider the potential inside (and on the surface) of the potential to be 0 and we plug r=R into the original equation $A_l R^{l}+B_l R^{-l-1}=0$ the rest of the boundary condition would be the same, which means we would get two different answers for the boundary condition that doesn't seem to contradict each other.

 

[bubblewrap]

See http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinates.html
I think you are ignoring the possibility of an $l=0$ solution. $\\$ You then have a separate solution inside the sphere that says $V=0$, that has both $A$ and $B =0$. $\\$ The coefficients of $A$ and $B$ inside the sphere are different from what they are outside. The potential $V$ is continuous across the boundary, but that doesn't mean that each of $A$ and $B$ stay the same. $\\$ Your first equation is correct, but for this case ($r \geq R$) you only have the $l=0$ term. I'm not sure how you came up with your second equation. $\\$ You really need to label the coefficients $A_{l \, out}$, $B_{l \, out}$, $A_{l \, in}$, and $B_{l \, in}$. $\\$ Edit: I looked at it a second time=I misread the problem. I thought $V=0$ everywhere at $r=R$ . Let me try again... It would help if you could show us the previous page, because otherwise it is a guessing game of what we are trying to solve.

But in the textbook it said that there are no terms $A_{l \, out}$ and $B_{l \, in}$, since they diverge at the points relevant to them.

 

[Charles Link]

But in the textbook it said that there are no terms $A_{l \, out}$ and $B_{l \, in}$, since they diverge at the points relevant to them.

I can believe that, but if I don't see the previous page, I don't know exactly what the problem consists of. $\\$ e.g. What is the constant $C$? I shouldn't need to guess what they are referring to.

 

[bubblewrap]

I can believe that, but if I don't see the previous page, I don't know exactly what the problem consists of. $\\$ e.g. What is the constant $C$? I shouldn't need to guess what they are referring to.

The constant is from the equation

$V=-E_0 z+C$ for $r>>R$

And the question was for a uncharged metal sphere of radius R that was placed in a electric field $E_0$ (which was taken to be the direction of z), find the potential outside the metal sphere. (Since the induced charges inside the sphere ball would distort the field/potential around it)

 

[Charles Link]

You might want to try a different source and compare what they both get for the same problem. The other source may also have an improved explanation. Try https://nptel.ac.in/courses/115101005/downloads/lectures-doc/Lecture-16.pdf
At the top of page 2 he talks about $B_o =0$ unless there is a charge on the sphere, as he mentions in the last sentence at the bottom of page 2.