(a) With the aid of ray diagrams, explain total internal reflection. (b) Describe, with the aid of a labelled diagram, the essential features of an astronom...

(a) Total internal reflection

Total internal reflection occurs when light travels from an optically denser medium to a less dense medium and the angle of incidence is greater than the critical angle, C. At i = C, the angle of refraction is \(90^\circ\), so that the refracted ray travels along the boundary. For \(i>C\), the ray is completely reflected back into the denser medium.

(b) Astronomical telescope in normal adjustment

An astronomical telescope consists of two converging lenses mounted on the same principal axis:

• the objective lens, of long focal length \(f_o\);
• the eyepiece lens, of short focal length \(f_e\).

Light from a distant object enters the objective as parallel rays. The objective forms a real, inverted and diminished intermediate image at its principal focus. In normal adjustment, this image is at the first focal point of the eyepiece, so that the principal foci coincide. The eyepiece then produces a final virtual image at infinity, and the emergent rays are parallel.

Hence, for normal adjustment,

\[d=f_o+f_e\]

where \(d\) is the separation of the lenses. The angular magnifying power has magnitude

\[M=\frac{f_o}{f_e}.\]

(c)

For a real image formed by a converging lens,

\[m=\frac{v}{u}=2\]

Therefore,

\[v=2u\]

Since the object and real image are on opposite sides of the lens, their separation is \(u+v\):

\[u+v=90.0\]

\[u+2u=90.0\]

\[3u=90.0\]

\[u=30.0\text{ cm},\qquad v=60.0\text{ cm}.\]

(i) Focal length

Using \(\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\),

\[\frac{1}{f}=\frac{1}{30.0}+\frac{1}{60.0}=\frac{3}{60.0}\]

\[\boxed{f=20.0\text{ cm}}\]

(ii) Object distance for an image of the same size

An image is the same size as the object when the object is at \(2f\).

\[u=2f=2(20.0)=\boxed{40.0\text{ cm}}\]