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Q.(a) Draw a labelled ray diagram showing the formation of an image by an astronomical refracting telescope in normal adjustment. Hence, obtain the expression for its magnifying power.

(OR)
(b) A plane wavefront of light of wavelength 'λ\lambda' is incident normally on a narrow slit of width 'aa' and a diffraction pattern is observed on a screen at a distance 'DD' from the slit.
(i) Depict the intensity distribution in the pattern observed.
(ii) Obtain the expression for the first maximum from the central maximum.
CBSECBSE Class XII Board 2022Subjective· 3mImportance★★★★★
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Part (a): For an astronomical telescope in normal adjustment, M=fo/feM=f_o/f_e, from the ratio of the angles subtended by image and object.

Part (b): Single-slit diffraction gives an (sin⁡β/β)2\left(\sin\beta/\beta\right)^2 intensity with a bright central maximum; the first secondary maximum is at sin⁡θ≈3λ/2a\sin\theta\approx 3\lambda/2a (not λ/a\lambda/a, which is the first minimum).

Ray diagram of an astronomical refracting telescope in normal adjustment: parallel rays from a distant object converge at the objective's focal point Fo to form a real, inverted intermediate image, which coincides with the eyepiece's focal point Fe so that rays emerge from the eyepiece parallel, forming the final image at infinity for a relaxed eye.
Ray diagram of an astronomical refracting telescope in normal adjustment: parallel rays from a distant object converge at the objective's focal point Fo to form a real, inverted intermediate image, which coincides with the eyepiece's focal point Fe so that rays emerge from the eyepiece parallel, forming the final image at infinity for a relaxed eye.

Part (a)

A telescope increases the angle subtended by a distant object. In normal adjustment the final image is at infinity (relaxed eye), so the tube length is L=fo+feL=f_o+f_e.

  • Ray diagram (described): parallel rays from a distant object enter the objective and converge to a real, inverted image A′B′A'B' at Fo′F_o'. The eyepiece is set so A′B′A'B' is at its focus FeF_e; rays then leave the eyepiece parallel and form a magnified virtual image at infinity.
  • Angles: the object subtends α\alpha at the objective; the final image subtends β\beta at the eye. With intermediate image height h=A′B′h=A'B':

tan⁡α≈α=hfo,tan⁡β≈β=hfe.\tan\alpha\approx\alpha=\frac{h}{f_o},\qquad \tan\beta\approx\beta=\frac{h}{f_e}.

  • Magnifying power: …

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