Find angular momentum of EM field in terms of q and ##\Phi##

In summary: Unfortunately, you have not yet shown that ##\int_0^{\pi} \Phi(\theta) \, \sin^3 \theta \, d\theta## can be simplified to ##\Phi##. In fact, it can't.Here is a hint on how to proceed. Consider the ##z##-component of the magnetic field at the point ##(x, y, z)## and the point ##(x, y, -z)##. How do these two components compare? What does this tell you about the magnitude of ##\mathbf l## at these two points? What does this tell you about the integration over all space of the z-component of ##\mathbf
  • #1
WeiShan Ng
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2

Homework Statement


A point charge q sits at the origin. A magnetic field ##\mathbf{B} (\mathbf{r})=B(x,y)\mathbf{\hat{z}}## fills all of space. The problem asks us to write down an expression for the total electromagnetic field angular momentum ##\bf{L_{EM}}##, in terms of q and the magnetic flux, ##\Phi_B## through the xy-plane

Homework Equations


momentum density
$$\mathbf{g} = \epsilon_0 (\mathbf{E} \times \mathbf{B})$$
angular momentum density:
$$l = \vec{r} \times \vec{g} = \epsilon_0 [\vec{r}\times (\vec{E}\times \vec{B})]$$

The Attempt at a Solution


The electric field from q is
$$\vec{E}=\frac{Q}{4\pi \epsilon_0 r^2} \mathbf{\hat{r}}$$
The magnetic field is
$$\vec{B}=B(x,y) \mathbf{\hat{z}}$$
So the angular momentum density is
$$\begin{align*} \mathbf{l} &= \epsilon_0 [\vec{r} \times (\mathbf{E}\times \mathbf{B})] \\
&=\epsilon_0 r\frac{QB}{4\pi \epsilon_0 r^2} [\mathbf{\hat{r} \times \hat{r} \times \mathbf{\hat{z}}}] \end{align*} $$
and
$$[\mathbf{\hat{r}\times \hat{r} \times \hat{z}}] = -r \sin \theta \cos \theta \cos \phi \mathbf{\hat{x}}+ r \sin \theta \cos \theta \sin \phi \mathbf{\hat{y}} + -r \sin^2 \theta \mathbf{\hat{z}}$$
The total angular momentum will be
$$\mathbf{L} = \iiint \mathbf{l} \, r^2 \sin \theta \, dr d\theta d\phi$$
Since we know that ##\int \sin \phi \, d\phi ## and ##\int \cos \phi \, d\phi## equal to zero in the interval ##[0, 2\pi]##, we can disregard x and y component, so the total angular momentum will be
$$\begin{align*} \mathbf{L} &= \iiint \frac{QB}{4\pi r} (-r \sin^2 \theta ) \, r^2 \sin \theta \, dr d\theta d\phi \, \mathbf{\hat{z}} \\ &= \iiint -\frac{QBr^2}{4\pi} \sin^3 \theta \, dr d\theta d\phi \end{align*}$$
Since we are considering xy-plane, (The surface element in a surface of polar angle θ constant) the magnetic flux can be written as
$$d\Phi = BdA = B r \sin \theta d\phi dr$$
$$\Phi = \iint B r \sin \theta d\phi dr$$
My problem is how can I substitute ##\Phi## into the equation of ##\mathbf{L}##, since I will have an extra ##r \sin^2 \theta ## in my integral ?
$$\mathbf{L} = \iiint -(B r \sin \theta) \frac{Qr\sin^2 \theta}{4\pi} \, dr d\theta d\phi \, \mathbf{\hat{z}}$$
 
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  • #2
Alright I found my own errors, turns out ##[\mathbf{\hat{r}\times \hat{r}\times \hat{z}}] = -\sin \theta \cos \theta \cos \phi \mathbf{\hat{x}}+ \sin \theta \cos \theta \sin \phi \mathbf{\hat{y}} + - \sin^2 \theta \mathbf{\hat{z}}## and since ##\theta = \pi/2## in xy plane, I can write $$\Phi = \iint B r d\phi dr$$ ...
 
  • #3
WeiShan Ng said:
So the angular momentum density is
$$\begin{align*} \mathbf{l} &= \epsilon_0 [\vec{r} \times (\mathbf{E}\times \mathbf{B})] \\
&=\epsilon_0 r\frac{QB}{4\pi \epsilon_0 r^2} [\mathbf{\hat{r} \times \hat{r} \times \mathbf{\hat{z}}}] \end{align*} $$
OK, except the triple cross product in the last line needs parentheses: ##[\mathbf{\hat{r} \times \left(\hat{r} \times \mathbf{\hat{z}}\right)}]##

and
$$[\mathbf{\hat{r}\times \hat{r} \times \hat{z}}] = -r \sin \theta \cos \theta \cos \phi \mathbf{\hat{x}}+ r \sin \theta \cos \theta \sin \phi \mathbf{\hat{y}} + -r \sin^2 \theta \mathbf{\hat{z}}$$
I believe the first term on the right should not have the negative sign. Also, the factor of ##r## in each term should not be there (as you noted in your post #2).

The total angular momentum will be
$$\mathbf{L} = \iiint \mathbf{l} \, r^2 \sin \theta \, dr d\theta d\phi$$
Since we know that ##\int \sin \phi \, d\phi ## and ##\int \cos \phi \, d\phi## equal to zero in the interval ##[0, 2\pi]##, we can disregard x and y component
This is not the reason that the x and y components can be neglected. Note that ##B(x,y)## is an arbitrary function of ##x## and ##y##. So, in spherical coordinates ##B## would be some function of ##r, \theta## and ##\phi##. Thus, instead of ##\int \sin \phi \, d\phi ##, you would need to consider ##\int B \sin \phi \, d\phi ## and this is not necessarily zero. Similarly, ##\int B \cos \phi \, d\phi ## is not necessarily zero. Try to find the correct reason why the the x and y components can be neglected. Hint: Consider the magnitude and direction of the angular momentum density ##\mathbf l## at points ##(x, y, z)## and ##(x, y, -z)##.

Since we are considering xy-plane, ...
You are not just considering the xy-plane. You must consider the integration to be over all of 3D space and show that this integration can be expressed in terms of ##Q## and the total magnetic flux ##\Phi## through the xy-plane.
 
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  • #4
I try to walk through the steps to see where I have done wrong:
TSny said:
ry to find the correct reason why the the x and y components can be neglected. Hint: Consider the magnitude and direction of the angular momentum density ll\mathbf l at points (x,y,z)(x,y,z)(x, y, z) and (x,y,−z)(x,y,−z)(x, y, -z).
Writing the cross products in Cartesian coordinates I have $$[\mathbf{\hat{r}\times (\hat{r}\times \hat{z})}] = \frac{xz}{r^2} \mathbf{\hat{x}} + \frac{yz}{r^2} \mathbf{\hat{y}}+\frac{x^2-y^2}{r^2} \mathbf{\hat{z}}$$ where ##r^2 = x^2+y^2+z^2## So the angular momentum density, ##\mathbf{l}## will be
$$\begin{align*}\mathbf{l} &= \frac{QB(x,y)}{4\pi r^3 }(xz \mathbf{\hat{x}}+ yz\mathbf{\hat{y}}+(x^2-y^2)\mathbf{\hat{z}}) \\ &= \frac{QB(x,y)}{4\pi (x^2+y^2+z^2)^{3/2} }(xz \mathbf{\hat{x}}+ yz\mathbf{\hat{y}}+(x^2-y^2)\mathbf{\hat{z}}) \end{align*}$$
x-component and y-component of ##\mathbf{l}## will have the same magnitude but opposite direction at ##\mathbf{l}(x,y,z)## and ##\mathbf{l}(x,y,-z)## for ##\forall (x,y)##. If we integrate it over all volume, they will cancel out and we only left with z-component.
TSny said:
You are not just considering the xy-plane. You must consider the integration to be over all of 3D space and show that this integration can be expressed in terms of QQQ and the total magnetic flux ΦΦ\Phi through the xy-plane.
The ##\Phi_B## is across xy-plane at z = 0, so it can be written as
$$d\Phi = BdA = B r \sin \theta d\phi dr$$ $$\Phi(\theta) = \iint B(r,\theta,\phi) r \sin \theta d\phi dr = \int_{-\infty}^\infty \int^{2\pi}_0 B(r,\theta,\phi) r d\phi dr $$ since ##\theta = \pi/2## at z=0
Integrate ##\mathbf{l}## over all space
$$\begin{align*}\mathbf{L} &= \iiint_{all space} \mathbf{l} \, r^2 \sin \theta d\phi dr d\theta \\
&= \int^{\pi}_0 \int^\infty_{-\infty} \int^{2\pi}_0 \frac{-QB \sin^2 \theta}{4\pi r} \mathbf{\hat{z}} \, r^2 \sin \theta d\phi dr d\theta \\
&=\int_0^{\pi} \left[ \int_{-\infty}^{\infty} \int_0^\pi \frac{-QB(r,\theta,\phi)r}{4\pi} d\phi dr \right] \sin^3 \theta d\theta \mathbf{\hat{z}} \\
&= \int_0^{\pi} \frac{-Q\Phi(\theta)}{4\pi} \sin^3 \theta d\theta \mathbf{\hat{z}} \end{align*}$$
And this will be my expression for ##\mathbf{L}##. Hopefully I get it right this time...
 
  • #5
WeiShan Ng said:
Writing the cross products in Cartesian coordinates I have $$[\mathbf{\hat{r}\times (\hat{r}\times \hat{z})}] = \frac{xz}{r^2} \mathbf{\hat{x}} + \frac{yz}{r^2} \mathbf{\hat{y}}+\frac{x^2-y^2}{r^2} \mathbf{\hat{z}}$$ where ##r^2 = x^2+y^2+z^2##
There is a sign error in the z-component term. But this will not affect your argumment for showing that the x and y components of ##\mathbf{L}_z## are zero.
So the angular momentum density, ##\mathbf{l}## will be
$$\begin{align*}\mathbf{l} &= \frac{QB(x,y)}{4\pi r^3 }(xz \mathbf{\hat{x}}+ yz\mathbf{\hat{y}}+(x^2-y^2)\mathbf{\hat{z}}) \\ &= \frac{QB(x,y)}{4\pi (x^2+y^2+z^2)^{3/2} }(xz \mathbf{\hat{x}}+ yz\mathbf{\hat{y}}+(x^2-y^2)\mathbf{\hat{z}}) \end{align*}$$
x-component and y-component of ##\mathbf{l}## will have the same magnitude but opposite direction at ##\mathbf{l}(x,y,z)## and ##\mathbf{l}(x,y,-z)## for ##\forall (x,y)##. If we integrate it over all volume, they will cancel out and we only left with z-component.
Yes. Nice.

The ##\Phi_B## is across xy-plane at z = 0, so it can be written as
$$d\Phi = BdA = B r \sin \theta d\phi dr$$
This is only valid for ##\theta = \pi/2##

$$\Phi(\theta) = \iint B(r,\theta,\phi) r \sin \theta d\phi dr $$
This will be the flux through the xy-plane only when ##\theta = \pi/2##. I don't think spherical coordinates are going to be convenient for this problem. You might try sticking with Cartesian coordinates.
 
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  • #6
TSny said:
There is a sign error in the z-component term. But this will not affect your argumment for showing that the x and y components of ##\mathbf{L}_z## are zero.
Yes. Nice.

This is only valid for ##\theta = \pi/2##

This will be the flux through the xy-plane only when ##\theta = \pi/2##. I don't think spherical coordinates are going to be convenient for this problem. You might try sticking with Cartesian coordinates.
Thanks for your help! I will try to redo the question in Cartesian coordinates.
 

Related to Find angular momentum of EM field in terms of q and ##\Phi##

1. What is angular momentum of an electromagnetic field?

The angular momentum of an electromagnetic (EM) field is a measure of the rotation and angular velocity of the field. It is a property of the electromagnetic field that is conserved in a closed system.

2. How is the angular momentum of an EM field calculated?

The angular momentum of an EM field is calculated by taking the cross product of the electric and magnetic field vectors, multiplied by the distance from the origin. It can also be calculated in terms of the charge (q) and magnetic flux (Φ) by using the formula L = qΦ.

3. What is the significance of the charge and magnetic flux in the calculation of angular momentum?

The charge (q) and magnetic flux (Φ) are important factors in determining the angular momentum of an EM field because they represent the amount of electric and magnetic energy present in the field. Together, they contribute to the rotational energy and angular momentum of the field.

4. How is angular momentum of an EM field related to its polarization?

The angular momentum of an EM field is related to its polarization because polarization is a measure of the direction and orientation of the electric field vector. This direction and orientation contribute to the rotational motion of the field, thus affecting its angular momentum.

5. What are some real-world applications of understanding the angular momentum of EM fields?

Understanding the angular momentum of EM fields is important in a variety of fields, including physics, engineering, and astronomy. It is used in the design of electric motors and generators, as well as in the study of celestial bodies and their magnetic fields. It also has implications in the development of renewable energy technologies.

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