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$(1)$ $|\Delta \phi|$ remains the same if the parabolic wire is replaced by a straight wire, $y=x$ initially, of length $\sqrt{2} L$
$(2)$ $|\Delta \phi|$ is proportional to the length of the wire projected on the $y$-axis.
$(3)$ $|\Delta \phi|=\frac{1}{2} B _0 V _0 L$ for $\beta=0$
$(4)$ $|\Delta \phi|=\frac{4}{3} B_0 V_0 L$ for $\beta=2$
$(i)$ Light can be totally internally reflected when it passes from $B$ to $A$
$(ii)$ Light can be totally internally reflected when it passes from $A$ to $B$
$(iii)$ Critical angle for total internal reflection is ${i_B} - {i_A}$
$(iv)$ Critical angle between $A$ and $B$ is ${\sin ^{ - 1}}\left( {\frac{{\sin {i_A}}}{{\sin {i_B}}}} \right)$