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Physics Question on Ray optics and optical instruments

In the measurement of the angle of a prism using a spectrometer, the readings of first reflected image are Vernier I:320∘40′;I : 320^\circ 40'; Vernier II:140∘30′II : 140^\circ 30' and those of the second reflected image are Vernier I:80∘38′;I : 80^\circ 38'; Vernier II:260∘24′. II : 260^\circ 24'. Then the angle of the prism is

A

59∘58′59^{\circ} 58'

B

59∘56′59^{\circ} 56'

C

60∘2′60^{\circ} 2'

D

60∘4′60^{\circ} 4'

Answer

59∘58′59^{\circ} 58'

Explanation

Solution

Angle of the prism,
A=ϕ2A = \frac{\phi}{2}
Now ϕ=ϕ1+ϕ22\phi = \frac{\phi_1 +\phi_2}{2}
Where ϕ1=x2−x1 \phi_1 = x_2 - x_1
=(360∘+80∘38′)−(320∘40′)= (360^\circ + 80^\circ 38')-(320^\circ 40')
=119∘58′= 119^\circ 58'
and ϕ2=y2−y1\phi_2 = y_2 - y_1
=260∘24′−140∘30′= 260^\circ 24' - 140^\circ 30'
=119∘54′= 119^\circ 54'
ϕ=119∘58′+119∘54′2\phi = \frac{119^\circ 58'+ 119^\circ 54'}{2}
=239∘52′2= \frac{239^\circ 52'}{2}
A=ϕ2=239∘52′4=59∘58′A =\frac{\phi }{2}= \frac{239^\circ 52'}{4}=59^\circ 58'