Check, with the use of (8), that the condition (7) is a...

Check, with the use of (8), that the condition (7) is a condition for minimal value of $t(a)$. From Fig. 5 determine which speed is more $-v_a$ or $v_b$ ? Using the formula (9), find out at what angles the light beam does not penetrate from medium $a$ into medium $b$, i.e. when the effect "of complete internal reflection of light" used in a number of technical devices is realized.

Key Concepts

Total Internal Reflection and Critical Angle

Total internal reflection occurs when a light beam, traveling from a medium with a higher speed (or lower refractive index) to one with a lower speed (or higher refractive index), hits the interface at an angle greater than a certain critical angle. At this point, the light does not penetrate into the second medium but is completely reflected back into the first medium. Determining this angle often involves the use of Snell's Law to relate the incident angle to the refractive indices of the two media.

Snell's Law

Snell's Law provides the relation between the angles of incidence and refraction when a light ray passes from one medium to another with different refractive indices, which are proportional to the inverse of the light speeds in the respective media. This law is crucial for understanding how light bends at interfaces and for comparing the speeds in the two media.

Stationary (or Extremum) Condition in Calculus of Variations

In many physical problems, including the path light takes, the condition for an extremum (often minimum) is that the derivative of the function (in this case, the travel time) with respect to a parameter is zero. This stationary condition is central to determining the optimal path a system follows, such as ensuring that t(a) is minimized.

Fermat's Principle

This principle states that light follows the path that minimizes (or more generally makes stationary) the optical path length, or equivalently, the time taken. It forms the basis for deriving laws of reflection and refraction by setting the derivative of the time function with respect to a variable to zero, thereby finding the path for which the time is at an extremum (usually a minimum).

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