Calculating equipotential surfaces

[Pengwuino]

So this may be a bit silly, but one thing I've never really learned in all my years is how one actually goes about calculating equipotential surfaces for arbitrary potentials? Let's say I have a potential that goes like

$$\Phi = A_0e^{-\left({{r}\over{r_0}}\right)^2}\cos^2 (\phi) \sin^2(\theta)$$

where $r, \phi$ are your typical spherical coordinates and $A_0$ is a constant and $r_0$ is there for dimensional purposes and I want to determine even a single equipotential surface, how does one go about doing this? You typically see very simple examples in physics and you're typically finding something trivial like the shells you see around point particles, but this relies on plain ol intuition. How does one determine the surface with arbitrary (but well-behaved) functions such as my example?

Edit: I changed my function a few times so that it's Laplacian has no weird features for what I'm doing.

 

[BruceW]

I'd suggest doing the gradient in spherical coordinates on the potential. This would give the normal to the equipotential surface at all points in space. Then just specify a particular value for the potential to get one particular surface.

 

[Pengwuino]

Hmm but that doesn't actually give me the surface as far as I can tell. I assume however one can define the surface, it'll be in terms of some parameter or relationship between the other coordinates.

My initial hunch would be you would set $\Phi = \Phi_0$ for generality, then determine the relationship $\phi = \phi(r,\theta)$ and my surface would be defined running through the $0 \le \phi < 2\pi$ and $0 \le \theta < \pi$.

I eventually have to figure out a surface integral which makes me wonder how do I even define a differential area element.

 

[Bill_K]

The total differential of Φ is dΦ = ∂Φ/∂x dx + ∂Φ/∂y dy + ∂Φ/∂z dz. To get an equipotential surface, set this equal to zero and solve for dz:

dz = - (∂Φ/∂x dx + ∂Φ/∂y dy)/(∂Φ/∂z)

This gives you a pair of PDEs to solve for the surface expressed as z(x,y):

∂z/∂x = - (∂Φ/∂x)/(∂Φ/∂z)
∂z/∂y = - (∂Φ/∂y)/(∂Φ/∂z)

 

[BruceW]

I eventually have to figure out a surface integral which makes me wonder how do I even define a differential area element.

So you want to integrate some other function (lets say f) over a surface defined such that the function $\Phi$ is kept constant?

 

[Pengwuino]

So you want to integrate some other function (lets say f) over a surface defined such that the function $\Phi$ is kept constant?

Yes, the function 'f' is actually going to be the gradient of $\Phi$.

So what I really need to figure out is given a potential, how can I define equipotential surfaces for it so that I can later integrate the gradient of the potential over those equipotential surfaces.

And to be clear, I would be integrating $\nabla \Phi \cdot d\vec{A}$

 

[BruceW]

$$\nabla \Phi \cdot d \vec{A} = \nabla^2 \Phi dV$$
But the Laplacian is zero for your problem , so luckily for you the integral is zero.

If $\Phi$ was not Laplacian, then Bill_K gives the method for finding the change in z when you change x or y (while still keeping on the equipotential surface). So this effectively defines an equipotential surface when you start from a particular value. To use this, you'd need to redefine your problem in cartesian coordinates.
Alternatively, you could do the total differential in spherical coordinates, which is more time-consuming, but you could get r as dependent on polar angle and azimuthal angle, which is more convenient for your problem.
This gives the 'parameter relationship' you were looking for. Then to integrate over this surface, you'd have to define the surface as being equal to the equipotential surface which you have just defined. I'm not sure how you'd do this. Maybe by creating a coordinate system where one variable is constant over the equipotential surface
i.e. you'd need to define one of your variables as equal to $\Phi$, and then integrate over the other two variables?

 

[Born2bwire]

$$\nabla \Phi \cdot d \vec{A} = \nabla^2 \Phi dV$$
But the Laplacian is zero for your problem , so luckily for you the integral is zero.

If $\Phi$ was not Laplacian, then Bill_K gives the method for finding the change in z when you change x or y (while still keeping on the equipotential surface). So this effectively defines an equipotential surface when you start from a particular value. To use this, you'd need to redefine your problem in cartesian coordinates.
Alternatively, you could do the total differential in spherical coordinates, which is more time-consuming, but you could get r as dependent on polar angle and azimuthal angle, which is more convenient for your problem.
This gives the 'parameter relationship' you were looking for. Then to integrate over this surface, you'd have to define the surface as being equal to the equipotential surface which you have just defined. I'm not sure how you'd do this. Maybe by creating a coordinate system where one variable is constant over the equipotential surface
i.e. you'd need to define one of your variables as equal to $\Phi$, and then integrate over the other two variables?

I don't think that's quite right. While the value of the Laplacian will be zero along the boundary of the integration (since it's equipotential) that doesn't mean that the interior value so the Laplacian will be zero. In fact, it should just work out to be a constant times the area of the equipotential surface.

Take for example the function,

$$\Phi(r,\theta,\phi) = r$$
This function has spherical shells of equipotential surfaces. Thus,

$$\nabla\Phi = \hat{r}$$
$$\nabla\Phi\cdot d\mathbf{S} = \hat{r}\cdot\hat{r} r^2\sin\theta d\theta d\phi$$
Thus the value of the integral over the equipotential surface comes out to be 4 \pi r^2.

 

[BruceW]

Woops, sorry for some reason I thought that Pengwuino said in the OP that his potential $\Phi$ satisfied Laplace's equation. But he was just saying the Laplacian has no weird features. My mistake.

 

[Born2bwire]

Woops, sorry for some reason I thought that Pengwuino said in the OP that his potential $\Phi$ satisfied Laplace's equation. But he was just saying the Laplacian has no weird features. My mistake.

Yeah, you had me thinking that too for a second.

 

[Pengwuino]

I don't think that's quite right. While the value of the Laplacian will be zero along the boundary of the integration (since it's equipotential) that doesn't mean that the interior value so the Laplacian will be zero.

I don't think I agree with the first part. The value of the Laplacian doesn't have to be zero along the boundary of the integration. Something trivial like $1/r^2$ has a non-zero laplacian over the spherical shells that define that functions equipotential surfaces.

 

[BruceW]

I think I agree with Pengwuino. Surely the Laplacian isn't necessarily zero anywhere for a general potential?
Back to the main problem: since the potential isn't spherical, I would re-write it in terms of the cartesian coordinates. Also, the gradient of the potential will be perpendicular to the equipotential surface, so:
$$\nabla \Phi \ \cdot \ d \vec{A} = \mid \nabla \Phi \mid \ dA$$
Now, to find what the equipotential surface is, we can use Bill K's method:
$$\frac{ \partial z}{ \partial x} = - \frac{ \frac{ \partial \Phi }{ \partial x} }{ \frac{ \partial \Phi }{ \partial z } }$$
This gives us the partial derivative of z with respect to x. And we can use this to define the length integral which is perpendicular to y. ie
$$ds = \sqrt{1 + { \frac{ \partial z}{ \partial x} }^2 } \ dz$$
So the integral becomes:
$$\mid \nabla \Phi \mid \ \sqrt{1 + { \frac{ \partial z}{ \partial x} }^2 } \ dz \ dy$$
Someone please say if this is wrong, its been ages since I did this kind of maths, but I think its right

Edit: After you've done these steps, I think you'll need to put x in terms of y and z before you do the integration, since x doesn't get integrated over? You can get the dependence of x on y and z by using Bill K's method. Basically, calculate the total differential of the potential along the surface, which we know is zero, and then calculate the total differential for x, then use these, along with boundary conditions, to get x(y,z).

 

[Born2bwire]

I don't think I agree with the first part. The value of the Laplacian doesn't have to be zero along the boundary of the integration. Something trivial like $1/r^2$ has a non-zero laplacian over the spherical shells that define that functions equipotential surfaces.

Right. I say we just delete this thread and start over. :shiftyeyes:

 

[BruceW]

Right. I say we just delete this thread and start over. :shiftyeyes:

Haha, we both made unfounded assumptions to begin with. I blame hopeful thinking, trying to make the problem easier.

 

[Pengwuino]

Ok so I've worked with a couple of potential functions and determined various surfaces. Now how does one form the surface element $d\vec A$? Gah, I need to grab a good differential geometry text.

I have 2 things I need to do. 1) Determine the differential area element and eventually integrate this function over that surface and 2) determine the limits of integration for a corresponding volume integral. I wonder if Spivak would help in this? His intro text.

 

[Born2bwire]

Ok so I've worked with a couple of potential functions and determined various surfaces. Now how does one form the surface element $d\vec A$? Gah, I need to grab a good differential geometry text.

I have 2 things I need to do. 1) Determine the differential area element and eventually integrate this function over that surface and 2) determine the limits of integration for a corresponding volume integral. I wonder if Spivak would help in this? His intro text.

If you are still trying to integrate the gradient of your potential over an equipotential surface, then you can take a few shortcuts. Namely, the gradient of the potential will always be constant and point in the same direction as your surface normal. So the result of the integral is the area of your surface times some constant related to the gradient of your potential. So if you have a simple surface (sphere, cylinder, etc) then you do not really need to work the integral. Only if you have a complicated surface would you need to work the integral by virtue to just get the area. So your surface element depends upon your system of coordinates and the variables that you are integrating over.

For example, in cartesian coordinates then it is always something like $\hat{z}dxdy$. If it is more complicated, then you should try to parameterize your equation for the surface.

 

[Pengwuino]

I've decided to use the potential $A_0e^{-{{r}\over{r_0}}^2}rcos(\theta)$

So far, I've determined the equipotential surfaces are given by

$$x^2 + y^2 + {{z^2}\over{2}} = {{r_0^2ln(z)}\over{2}}+C$$

where the C is a constant that will denote the equipotential surface I believe. These surfaces are very complicated so I can't use any symmetry arguments to make things simpler. I wish I knew how to even look at the surface in Mathematica.

 

[BruceW]

If you are still trying to integrate the gradient of your potential over an equipotential surface, then you can take a few shortcuts. Namely, the gradient of the potential will always be constant and point in the same direction as your surface normal. So the result of the integral is the area of your surface times some constant related to the gradient of your potential.

The gradient of the potential would point in the direction of the surface normal, but why would the gradient of the potential be constant?

 

[Pengwuino]

The gradient of the potential would point in the direction of the surface normal, but why would the gradient of the potential be constant?

Yah that can't be true for arbitrary surfaces.

 

[Born2bwire]

The gradient of the potential would point in the direction of the surface normal, but why would the gradient of the potential be constant?

I feel ashamed with myself. I can't believe I've been making these posts. Ok... where is that memory erase and reset button for this thread...

 

[meldraft]

Actually, I think you are really overthinking here. Since you have an actual FUNCTION for the potential, if you just set the function equal to a constant, i.e. f=5, you can immediately get the equipotential surface (the equal potential being.. 5!)

 

[Pengwuino]

Actually, I think you are really overthinking here. Since you have an actual FUNCTION for the potential, if you just set the function equal to a constant, i.e. f=5, you can immediately get the equipotential surface (the equal potential being.. 5!)

That doesn't tell you anything about what the surface actually is. For a radial function, the surfaces are trivially to find, but for anything else not so much.

 

[meldraft]

Mmm I get what you mean. However, you can still plot it right? For instance, the equipotential lines for:

$$A_0e^{−\frac{r^2}{r_0}}rcos(θ)$$

are (I set A0=5 and r0=2):

 

[BruceW]

I've decided to use the potential $A_0e^{-{{r}\over{r_0}}^2}rcos(\theta)$

So far, I've determined the equipotential surfaces are given by

$$x^2 + y^2 + {{z^2}\over{2}} = {{r_0^2ln(z)}\over{2}}+C$$

where the C is a constant that will denote the equipotential surface I believe.

Wouldn't the equipotential surface be defined by:
$$e^{-{\frac{r}{r_0}}^2} \ rcos(\theta) = constant$$
Which can be rearranged as:
$${r_0}^2 \ ln(z) -x^2 -y^2 - z^2 = another \ constant$$

 

[Pengwuino]

oops yes, I messed up the calculation, what you have is correct.

 

[BruceW]

I think that what I said in post 12 about the integral is wrong, because z depends on y and x.
BUT I think that method will work if using spherical coordinates, since we can say r depends on only theta, not phi. So:
$$dS^2 = r^2 d \theta^2 + d r^2$$
rearranging (this step is only valid since r doesn't depend on phi):
$$dS = \sqrt{r^2 +{( \frac{dr}{d \theta})}^2} \ d \theta$$
And now integrating over phi:
$$dA = \sqrt{r^2 +{( \frac{dr}{d \theta})}^2} \ r sin( \theta) d \theta d \phi$$
But we need to put r in terms of theta (and we need to do that for the function we are integrating also). The equation for the equipotential surface tells us both $\frac{dr}{d \theta}$ and how to write one variable in terms of the other:
$$e^{ - { \frac{r}{r_0}}^2} r cos(\theta) = C$$
After suitable rearranging, you can get one in terms of the other.

But obviously when you try to work this out, the integral gets really difficult. I suggest ignoring all the stuff above, and instead use the relation:
$${\nabla}^2 \Phi \ dV = \nabla \Phi \ \cdot \ d \vec{A}$$
So that you can integrate over a volume (hopefully simpler).

 

[Pengwuino]

Actually the volume is difficult as well as far as I can tell, I'm not sure how one would go about doing the integration other than numerically.