Find potential within a pipe using Laplace's equation

[Bestphysics112]

Homework Statement

Hello,

I'm trying to solve laplaces equation to find a solution for the potential in a pipe with the given boundary conditions:

at x=b, V=V_0

at x= -b, V = -V_0

at y=a, V=0

at y=-a, V=0

(Assume this configuration is centered on the origin, pipe as dimensions -b<=x<=b -a<=x<=a, infinite in z direction)

I have general form of solution (with exponential and trig combinations, I found this in griffith's section on separation of variables). I'm stumped on how to obtain more information from the boundary conditions. Here is what I'm referring to

https://imgur.com/a/wpKAY

For example, If the potential is 0 at y=a (which we are given), what exactly does this tell us for the coefficients? In the text, the example deals with instances in which one of the boundaries is the origin (x and y at 0), so I can eliminate a coefficient. But with the boundary at a non zero x and y coordinate, what should I do? Please don't give the full solution, but please give me a hint of sorts. Sorry if my english is poor, If my question doesn't make sense please say so and I will try to fix

Homework Equations

https://imgur.com/a/wpKAY (Potential expression obtained for cartesian coordinates.)

The Attempt at a Solution

Ok, here is my attempt at the solution. For the last two boundary conditions we are given, we can find that coefficient D=-D or C=-C. I'm not sure how to proceed. I've done similar problems where at least two of the boundary conditions lie on the origin (i.e. at x = 0 or y = 0). If it was like this, it is easy to see that a certain coefficient will be 0 and so forth. If, after plugging in y = +- a, I get the mentioned expressions, how can I determine what value C and D are? I asked someone and they said that C (or D) is zero but how can I determine which is 0? Because surely they both cannot be 0. A hint would be greatly appreciated.

 

[haruspex]

For the last two boundary conditions we are given, we can find that coefficient D=-D or C=-C.

How do you deduce that? What trig function would be zero at -a and +a?

 

[Bestphysics112]

How do you deduce that? What trig function would be zero at -a and +a?

My reason is as follows - If I plug y = a into (Csin(ky)+Dcos(ky)), I can solve for C. Then if I plug in y = -a and my value for C, I can get -Dcos(ak) = Dcos(ak) (because sin is odd function and cos is even function) I'm not sure if my reasoning is correct so feedback is appreciated. I'm not sure which trig function would be zero at -a and +a, can I get a hint? Is it cos because cos(0) = 1 so cos at the boundaries would be 0?

 

[haruspex]

get -Dcos(ak) = Dcos(ak)

And from that you deduce D=-D? What is another possibility?

 

[Bestphysics112]

And from that you deduce D=-D? What is another possibility?

If I do the same but instead solve for C, I can find that C= -C.

 

[haruspex]

If I do the same but instead solve for C, I can find that C= -C.

That does not answer my question. Read it again.

 

[Bestphysics112]

That does not answer my question. Read it again.

Another possibility is D=0? I'm unsure of what else it could be

 

[haruspex]

Another possibility is D=0? I'm unsure of what else it could be

If xy=0, what are the possibilities?

 

[Bestphysics112]

If xy=0, what are the possibilities?

Either x = 0 or y = 0

Edit. I think I see what your point is. The other possibility would be that cos(ak) = 0, meaning k = n*pi/(2a)

 

[haruspex]

Either x = 0 or y = 0

Right. Apply that principle to the equation I quoted in post #4.

 

[Bestphysics112]

If xy=0, what are the possibilities?

Right. Apply that principle to the equation I quoted in post #4.

I think I see what your point is. The other possibility would be that cos(ak) = 0, meaning k = n*pi/(2a)

 

[haruspex]

I think I see what your point is. The other possibility would be that cos(ak) = 0, meaning k = n*pi/(2a)

Right.

 

[Bestphysics112]

Right.

How does this fact help me in determining which (C or Do) is 0? Sorry for all the questions!

 

[haruspex]

How does this fact help me in determining which (C or Do) is 0? Sorry for all the questions!

Why do you keep writing "C or D is zero"? For each you had an equation which (you thought) implied it was zero, so surely you thought you had shown they were both zero.
Now you have seen that D need not be zero. What about C?

 

[Bestphysics112]

Why do you keep writing "C or D is zero"? For each you had an equation which (you thought) implied it was zero, so surely you thought you had shown they were both zero.
Now you have seen that D need not be zero. What about C?

C need not be zero as well, so there must be another way to determine coefficients

 

[haruspex]

C need not be zero as well,

Please post you complete reasoning for that.

 

[Bestphysics112]

Please post you complete reasoning for that.

If I follow the steps illustrated in post 3 but instead solve for d, I obtain -Csin(ak)=Csin(ak), so two possibility is C=0 or sin(ak)=0, so k=n*pi/a for this instance

 

[Chestermiller]

Can you sketch what you think the solution is going to look like in terms of the lines of constant V?

 

[haruspex]

If I follow the steps illustrated in post 3 but instead solve for d, I obtain -Csin(ak)=Csin(ak), so two possibility is C=0 or sin(ak)=0, so k=n*pi/a for this instance

Still not seeing how you get that. Pleaee show all the steps.