I'm trying to follow a text I found online. The author decomposes EM fields such
$$ \mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d g_l(r)}{dr} + g_l(r) \right) \Phi_{lm}\right) e^{-i\omega t} $$ $$ \mathbf{B} = \sum_{lm}\left(g_l(r) \mathbf{Y}_{lm} + i \frac{l(l+1)}{r} f_l(r) \mathbf{\Psi}_{lm} + i\left(\frac{d f_l(r)}{dr} + f_l(r) \right) \Phi_{lm}\right)e^{-i\omega t} $$ Where the functions are defined as $$ \mathbf{Y}_{lm} = r \mathbf{n} \times \mathbf{\nabla} Y_{lm} $$ $$ \mathbf{\Psi}_{lm} = r \mathbf{\nabla} Y_{lm} $$ $$ \mathbf{Y}_{lm} = Y_{lm} \mathbf{n} $$ where $\mathbf{n}$ is the position vector $\mathbf{n} = \mathbf{r}/r$, $Y_{lm}$ are the vector spherical harmonics (the arguments $\theta, \phi$ are omitted), $f_l(r),g_l(r)$ are either the spherical Bessel functions or the spherical Bessel functions dependent only on the radial coordinate and $\mathbf{\nabla}$ is the gradient.
Now the author states, that using the equations $$ \mathbf{n} \times \mathbf{E} = 0 $$ $$ \mathbf{n} \cdot \mathbf{B} = 0 $$ and by utilizing the spherical harmonics 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.
