No $B$, Speed Alone

Why does total internal reflection occur?

From the post "Wave Front and Wave Back" dated 18 May 2014, a photon was conceptualized as a particle in helical motion,

$\cfrac{x_{v1}}{cos(\alpha_1)}=\cfrac{x_{v2}}{cos(\alpha_2)}$

$x_{v2}=x_{v1}\cfrac{cos(\alpha_2)}{cos(\alpha_1)}$

where $x_{v1}$,  $x_{v2}$ are the radii of circular motion in medium 1 and 2 respectively.

and

$\cfrac{\lambda}{n_1sin(\alpha_1)}=\cfrac{\lambda}{n_2sin(\alpha_2)}$

$sin(\alpha_2)=\cfrac{n_1}{n_2}sin(\alpha_1)$

So,

$x_{v2}=\cfrac{x_{v1}}{cos(\alpha_1)}\sqrt{1-\left(\cfrac{n_1}{n_2}\right)^2sin^2(\alpha_1)}$

when the particle enters into to less dense medium,

$n_2\lt n_1$

$1-\left(\cfrac{n_1}{n_2}\right)^2sin^2(\alpha_1)\lt 0$

in which case, $x_{v2}$ is complex and is rotated by $90^o$ clockwise at the point of ncident,

$x_{v2}=i.\cfrac{x_{v1}}{cos(\alpha_1)}\sqrt{\left|1-\left(\cfrac{n_1}{n_2}\right)^2sin^2(\alpha_1)\right|}$

and $\alpha_2$ is totally internally reflected.  When

$1-\left(\cfrac{n_1}{n_2}\right)^2sin^2(\alpha_1)=0$

$sin(\alpha_1)=sin(\alpha_c)=\cfrac{n_2}{n_1}$

where $\alpha_c$ is the critical angle.  Unfortunately, the formula is valid only up to $\alpha_c$.  For incident angle greater than $\alpha_c$, we know that the ray is reflected,

$x_{v2}={x_{v1}}$

$1=\cfrac{cos(\alpha_2)}{cos(\alpha_1)}$

$\alpha_1=\alpha_2$

both angles measured from the normal on medium $n_1$.

This derivation for total internal reflection considers the relative speeds of the particle in the two mediums alone; $B$ fields are not involved.  Since, both loops are perpendicular to the ray $\alpha$ only in the limiting case of $\theta\to90^o$, the following adjustments are necessary to the values of $\alpha$ for each of the loop as illustrated,

$\alpha_{adj}=\alpha+90^o-\theta$

and

$\alpha_{adj}=\alpha-90^o+\theta$

which indicate that the two loops can be separated (circular polarization$\to$linear polarization) when,

since $\alpha\lt90^o$

$\alpha-90^o+\theta\lt\alpha_c$

$\alpha\lt\alpha_c+90^o-\theta$

and

$\alpha+90^o-\theta\gt\alpha_c$

$\alpha\gt\alpha_c-90^o+\theta$

where $\alpha_{2}^{'}$ has been totally internally reflected.  When $\theta\to90^o$, the range of $\alpha$ collapses to a single value $\alpha_c$, as $\alpha_{adj}\to\alpha$.