Calculating crossproduct integral, Parametrization

[Karl Karlsson]

Homework Statement

Consider a circular solenoid with length L and radius R. The solenoid is wound with n revolutions per unit length and is traversed by a current $I_0$. If the solenoid is tightly wound (ie, n is large), the current density can be approximated by a surface current of the form $d\vec I = I_0n\vec e_3\times d\vec S$.

The magnetic force $d\vec F$ from an external magnetic field $\vec B$ on a small surface element is given by $d\vec F = d\vec I\times \vec B$

i) Motivate and write down a parameterization of the surface occupied by the solenoid. (Start by placing the solenoid in an appropriately selected way in a Cartesian coordinate system.) Specify in particular how the Cartesian coordinates depend on your parameters and which parameter values are allowed.

ii) Calculate the surface element $d\vec S$ in your parameterization.

iii) The solenoid is placed in an arbitrary constant magnetic field $\vec B$. Write down the expression for dF and simplify this as far as possible.

iiii) The total moment of force on the solenoid relative to a position $\vec x_0$ is obtained by integrating $\vec T = \int_S (\vec x - \vec x_0 )\times d\vec F$ over the entire solenoid surface S. Calculate this integral for the situation described in iii) when $\vec x_0$ is the center of the solenoid.

Relevant Equations

$d\vec I = I_0n\vec e_3\times d\vec S$
$d\vec F = d\vec I\times \vec B$
$\vec T = \int_S (\vec x - \vec x_0 )\times d\vec F$

i) I approximate the solenoid as a cylinder with height L and radius R. I am not sure how I am supposed to place the solenoid in the coordinate system but I think it must be like this, right?

The surface occupied by the cylinder can be described by all vectors $\vec x =(x,y,z)$ so that $x^2+y^2=R^2$ , 0 ≤ z ≤ L. One parametrization to this could be $x(t) = R\cdot cos(t), y(t)=R\cdot sin(t)$ , 0 ≤ z ≤ L , 0≤t ≤ $2\pi$ , t = 0 at the x-axis and positive rotation is counterclockvise around the z axis. $\vec x = (R\cdot cos(t), R\cdot sin(t), z)$

ii) I get $d\vec S = \frac {d\vec x} {dt}\times \frac {d\vec x} {dz} \cdot dzdt = (R\cdot cos(t), R\cdot sin(t), 0) \cdot dzdt$. Do you get the same and is this correct?

iii) From here the expressions get so long that I will post a picture with what I wrote by hand and I wrote ρ instead of t while doing the calculations below so t = ρ

Is there any faster way to get to the result above?

iiii) Here the calculations will be even longer. If I insert the expression i got in iii) above in the integral in iiii) as the picture below shows, then I get a long expression

So as you can see I only started calculating the vectorproduct within the integral above until I had a feeling this is not the fastest way of solving this problem and maybe iii) as well? But how else are you supposed to do it?

Thanks in advance!

 

[pasmith]

Cylindrical polar coordinates $(r, \phi, z)$ are the obvious choice. With an eye to part (iv) I would take $-\frac12L \leq z \leq \frac12L$ rather than $0 \leq z \leq L$ so that $\mathbf{x}_0 = 0$.

The surface element of the cylinder with outward normal is then $$d\mathbf{S} = R\mathbf{e}_r\,d\phi\,dz$$ where $$\begin{align*}\mathbf{e}_r &= \cos \phi\,\mathbf{e}_x + \sin \phi\,\mathbf{e}_y \qquad\mbox{and} \\ \mathbf{e}_\phi &= -\sin \phi\,\mathbf{e}_x + \cos \phi\,\mathbf{e}_y \end{align*}$$ are the unit vectors in the directions of increasing $r$ and $\phi$ respectively.

Now set $\mathbf{B} = B_r \mathbf{e}_r + B_\phi \mathbf{e}_\phi + B_z \mathbf{e}_z$ and calculate the cross products using the identities $$\mathbf{e}_r \times \mathbf{e}_\phi = \mathbf{e}_z, \qquad \mathbf{e}_\phi \times \mathbf{e}_z = \mathbf{e}_r, \qquad \mathbf{e}_z \times \mathbf{e}_r = \mathbf{e}_\phi.$$

 

[Karl Karlsson]

Cylindrical polar coordinates $(r, \phi, z)$ are the obvious choice. With an eye to part (iv) I would take $-\frac12L \leq z \leq \frac12L$ rather than $0 \leq z \leq L$ so that $\mathbf{x}_0 = 0$.

The surface element of the cylinder with outward normal is then $$d\mathbf{S} = R\mathbf{e}_r\,d\phi\,dz$$ where $$\begin{align*}\mathbf{e}_r &= \cos \phi\,\mathbf{e}_x + \sin \phi\,\mathbf{e}_y \qquad\mbox{and} \\ \mathbf{e}_\phi &= -\sin \phi\,\mathbf{e}_x + \cos \phi\,\mathbf{e}_y \end{align*}$$ are the unit vectors in the directions of increasing $r$ and $\phi$ respectively.

Now set $\mathbf{B} = B_r \mathbf{e}_r + B_\phi \mathbf{e}_\phi + B_z \mathbf{e}_z$ and calculate the cross products using the identities $$\mathbf{e}_r \times \mathbf{e}_\phi = \mathbf{e}_z, \qquad \mathbf{e}_\phi \times \mathbf{e}_z = \mathbf{e}_r, \qquad \mathbf{e}_z \times \mathbf{e}_r = \mathbf{e}_\phi.$$

Hi! Apperently the magnetic field is supposed to be constant. So I had to write B with cartesian coordinates for it to be constant. Is my attempt on iiii) below correct and is this the shortest way of solving iiii)? I assumed I had done iii) correctly. Is iii) correct?

Thanks in advance!

 

[pasmith]

I agree with you as far as $$\mathbf{T} = nI_0 R \int_{-L/2}^{L/2} \int_0^{2\pi} (RB_r +zB_z) \mathbf{e}_\phi\,d\phi\,dz.$$

Unfortunately at this point you have substituted the wrong value for $\mathbf{e}_\phi = -\sin\phi\,\mathbf{e}_x + \cos\phi\,\mathbf{e}_y$.

Also, because the solenoid is axially symmetric, you can choose the direction of the $x$ axis such that $\mathbf{B} = B_x\mathbf{e}_x + B_z\mathbf{e}_z$. This should simplify the calculations.

 

[Karl Karlsson]

I agree with you as far as $$\mathbf{T} = nI_0 R \int_{-L/2}^{L/2} \int_0^{2\pi} (RB_r +zB_z) \mathbf{e}_\phi\,d\phi\,dz.$$

Unfortunately at this point you have substituted the wrong value for $\mathbf{e}_\phi = -\sin\phi\,\mathbf{e}_x + \cos\phi\,\mathbf{e}_y$.

Thanks for spotting that misstake!

Also, because the solenoid is axially symmetric, you can choose the direction of the $x$ axis such that $\mathbf{B} = B_x\mathbf{e}_x + B_z\mathbf{e}_z$. This should simplify the calculations.

I did not think of that. That makes the expressions shorter, thanks!

 

[Karl Karlsson]

Could somebody please check if my solutions on i - iiii are correct in the pdf attached to this post?

Thanks in advance!