# Working with decomposition of fields

I'm trying to follow a text I found online. The author decomposes EM fields such

$$\mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d g_l(r)}{dr} + \frac{g_l(r)}{r} \right) \mathbf{\Phi}_{lm}\right) e^{-i\omega t}$$ $$\mathbf{B} = \sum_{lm}\left(g_l(r) \mathbf{Y}_{lm} + i \frac{l(l+1)}{r} f_l(r) \mathbf{\Psi}_{lm} + i\left(\frac{d f_l(r)}{dr} + \frac{f_l(r)}{r} \right) \mathbf{\Phi}_{lm}\right)e^{-i\omega t}$$ Where the functions are defined as $$\mathbf{\Phi}_{lm} = r \mathbf{\hat r} \times \mathbf{\nabla} Y_{lm}$$ $$\mathbf{\Psi}_{lm} = r \mathbf{\nabla} Y_{lm}$$ $$\mathbf{Y}_{lm} = Y_{lm} \mathbf{\hat r}$$ where $$\mathbf{\hat r}$$ is the position unit vector $$\mathbf{\hat r} = \mathbf{r}/r$$, $$Y_{lm}$$ are the vector spherical harmonics (the arguments $$\theta, \phi$$ are omnitted), $$f_l(r),g_l(r)$$ are "good behaving" radial functions and $$\mathbf{\nabla}$$ is the gradient.

Now the author states, that using the equations $$\mathbf{\hat r} \times \mathbf{E} = 0$$ $$\mathbf{\hat r} \cdot \mathbf{B} = 0$$ and by utilising the Spherical harmonic orthogonality we get $$f_l(r) = 0$$ $$\frac{d g_l(r)}{dr} = 0$$ But this doesn't work for me , I always get $$\frac{dg_l(r)}{dr} + r g_l(r) = 0$$ Could someone check the answer with me or prove me wrong? For further reading on the Functions $$(\mathbf{Y},\mathbf{\Psi},\mathbf{\Phi})$$ please refer to this text.

• Perhaps further elaboration will help motivate someone to answer. The problem is a boundary one, if we consider $f_l(r),g_l(r)$ to be Spherical Bessel/Hankel functions (althought with different coeficients) the equations for E and B represent EM waves. If we denote, now the equations $\mathbf{n} \times \mathbf{E}= 0$ and $\mathbf{n} \times B$ are the boundary conditions, if we solve the equations we should be able to find the coefficient for transmision/reflection of the waves. (note that we are missing some equations for the boundary, I have not included these as to not confuse readers. Apr 27, 2022 at 13:21
• You have expressions of the form $\frac{df}{dr}+f$ in your equations, which look dimensionally inconsistent. Are you sure these equations are correct? Apr 30, 2022 at 16:08
• Thank you, there should be, of course, $\frac{f}{r}$ May 1, 2022 at 13:25

Using: $$\mathbf{\hat r}\times \mathbf \Phi_{lm} = -\mathbf \Psi_{lm}$$ $$\mathbf{\hat r}\times \mathbf \Psi_{lm} = \mathbf \Phi_{lm}$$ $$\mathbf{\hat r}\times \mathbf Y_{lm} = 0$$ I get from $$\mathbf{\hat r}\times \mathbf E=0$$ at the boundary and the independence of the harmonic functions: $$\frac{g_l}{r}=\frac{dg_l}{dr}+\frac{g_l}{r}=0$$ which gives $$\frac{dg_l}{dr}=0$$.
Note that you second boundary condition, $$\mathbf{\hat r}\cdot \mathbf B = 0$$ using: $$\mathbf{\hat r}\cdot \mathbf \Phi_{lm} =\mathbf{\hat r}\cdot \mathbf \Psi_{lm} = 0$$ $$\mathbf{\hat r}\cdot \mathbf Y_{lm} = Y_{lm}$$ gives only $$g_l=0$$ at the boundary.
There is no condition on $$f_l$$ based on what you've given us. Are you sure you gathered all the information, and you don't have a typo? What is your reference?