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Question
A totally reflecting, small plane mirror placed horizontally faces a parallel beam of light, as shown in the figure. The mass of the mirror is 20 g. Assume that there is no absorption in the lens and that 30% of the light emitted by the source goes through the lens. Find the power of the source needed to support the weight of the mirror.
(Use h = 6.63 × 10-34J-s = 4.14 × 10-15 eV-s, c = 3 × 108 m/s and me = 9.1 × 10-31kg)
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Solution
Given :-
Mass of the mirror, m = 20 g = 20 × 10−3 kg
The weight of the mirror will be balanced if the force exerted by the photons will be equal to the weight of the mirror.
Now,
Relation between wavelength `(λ)` and momentum (p) :-
`p = h/λ`
On divinding both sides by t , we get
`P/t = h/(λt)...............(1)`
Energy,
`E = (hc)/λ`
⇒ `E/t = (hc)/(λt)`
Let P be the power. Then,
`P = E/t = (hc)/(λt)`
`P = (pc)/t ............ ["Using equation (1)"]`
⇒ `P/c = p/t`
Force ,
`F = P/t = P/c ..............("Since F" = ("Momentum")/("Time"))`
Thus, rate of change of momentum = Power/c
As the light gets reflected normally,
Force exerted = 2 (Rate of change of momentum) = 2 × Power/c
`30% "of" ((2 xx "Power")/c) = "mg"`
`⇒ "Power" = (20 xx 10^-3 xx 10 xx 3 xx 10^8 xx 10)/(2 xx 3)= 100" MW"`
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