# Orbital and Spin Angular momentum of light derivation

1. Sep 5, 2015

### calculo2718

1. The problem statement, all variables and given/known data
The momentum carried by an electromagnetic field is [;\vec{P}(\vec{x}, t) = \frac{1}{4\pi c} \int d\vec{x}\vec{E}(\vec{x},t) \times \vec{B}(\vec{x},t);]

show that for a finite field extension
[;\vec{J}(\vec{x}, t) = \frac{1}{4\pi c}\int -i A_i(\vec{x},t)\vec{L}E_i(\vec{x},t) + \vec{E}(\vec{x},t) \times \vec{A}(\vec{x},t) d\vec{x};] where [;\vec{J};] is the angular momentum, [;L = -ir \times \nabla;], and [;\vec{A};] is the vector potential in radiation gauge, i.e. [;A^0 = 0;] and [;\nabla \cdot \vec{A} = 0;] (hint: using partial integration)
2. Relevant equations
coulomb gauge [;\vec{B} = \nabla \times \vec{A};]

3. The attempt at a solution
There are some things in the wording in the problem that I do not understand so I am making some assumptions that may not even be true.

1) Finite field extension means that the field does NOT go to infinite so [;\nabla \cdot \vec{E} = 0;]
2) The subscripts in the [;-i A_i(\vec{x},t)\vec{L}E_i(\vec{x},t);] represents a sum.
3) I have no clue what partial integration is.
[;\int \vec{x} \times (\vec{E} \times \vec{B}) d\vec{x};]
[;\int \vec{x} \times (\vec{E} \times (\nabla \times \vec{A})) d\vec{x};]
using triple product expansion
[;\int \vec{x} \times (\nabla(\vec{E} \cdot \vec{A}) - \vec{A}(\vec{E}\cdot \nabla)) d\vec{x};]
distributing the [;\vec{x} \times;]
[;\int \vec{x} \times \nabla(\vec{E} \cdot \vec{A}) - \vec{x} \times \vec{A}(\vec{E}\cdot \nabla) d\vec{x};]

This is where I get stuck. If I assume that [;\nabla \cdot \vec{E} = 0;], I have no idea where to get the [;\vec{E} \times \vec{A};] term in the answer I am supposed to get. If I don't assume that, I try to expand my [;\nabla(\vec{E} \cdot \vec{A});] term using some vector identity the [;\nabla \cdot \vec{E};] cancels anyway and I simply get the equation in my second line of work.

My gut tells me that I missing something having to do with this "partial integration" business. I have googled this and I get it as an alternative term to integration by parts or as integrating a partial derivative. If I am to integrate by parts, what am I integrating by parts? I don't have a product of two functions I have different vector products of different vector fields. If I am supposed to integrate some partial derivative, which one? I have so many!

2. Sep 5, 2015

Your TeX formulas get displayed properly if you enclose them in $$or ## (inline mode). Currently they are hard to read. 3. Sep 5, 2015 ### calculo2718 I can't seem to edit the post so I am re-posting here. 1. The problem statement, all variables and given/known data The momentum carried by an electromagnetic field is$$\vec{P}(\vec{x}, t) = \frac{1}{4\pi c} \int d\vec{x}\vec{E}(\vec{x},t) \times \vec{B}(\vec{x},t)$$show that for a finite field extension$$\vec{J}(\vec{x}, t) = \frac{1}{4\pi c}\int -i A_i(\vec{x},t)\vec{L}E_i(\vec{x},t) + \vec{E}(\vec{x},t) \times \vec{A}(\vec{x},t) d\vec{x}] where $\vec{J}$$is the angular momentum,$$L = -ir \times \nabla$, and [\vec{A}$$is the vector potential in radiation gauge, i.e.$$A^0 = 0$$and$$\nabla \cdot \vec{A} = 0$$(hint: using partial integration) 2. Relevant equations coulomb gauge$$\vec{B} = \nabla \times \vec{A}$$3. The attempt at a solution There are some things in the wording in the problem that I do not understand so I am making some assumptions that may not even be true. 1) Finite field extension means that the field does NOT go to infinite so$$\nabla \cdot \vec{E} = 0$$2) The subscripts in the$$-i A_i(\vec{x},t)\vec{L}E_i(\vec{x},t)$$represents a sum. 3) I have no clue what partial integration is.$$\int \vec{x} \times (\vec{E} \times \vec{B}) d\vec{x}\int \vec{x} \times (\vec{E} \times (\nabla \times \vec{A})) d\vec{x}$$using triple product expansion$$\int \vec{x} \times (\nabla(\vec{E} \cdot \vec{A}) - \vec{A}(\vec{E}\cdot \nabla)) d\vec{x}$$distributing the$$\hat{x}\times\int \vec{x} \times \nabla(\vec{E} \cdot \vec{A}) - \vec{x} \times \vec{A}(\vec{E}\cdot \nabla) d\vec{x}$$This is where I get stuck. If I assume that$$\nabla \cdot \vec{E} = 0$$, I have no idea where to get the$$\vec{E} \times \vec{A}$$term in the answer I am supposed to get. If I don't assume that, I try to expand my$$\nabla(\vec{E} \cdot \vec{A})$$term using some vector identity the$$\nabla \cdot \vec{E} cancels anyway and I simply get the equation in my second line of work.

My gut tells me that I missing something having to do with this "partial integration" business. I have googled this and I get it as an alternative term to integration by parts or as integrating a partial derivative. If I am to integrate by parts, what am I integrating by parts? I don't have a product of two functions I have different vector products of different vector fields. If I am supposed to integrate some partial derivative, which one? I have so many!