\(R\) ત્રિજ્યા અને \(M\) દળ ધરાવતા કાળા કલરના ગોળા ની અંદર બખોલ છે જેની અંદર શૂન્યાવકાશ છે.બખોલની દીવાલનું તાપમાન \(T_0\) જાળવી રાખવામા આવેલ છે. ગોળાનું શરૂઆતનું તાપમાન \(3T_0\) છે.જો \(T\) તાપમાને રહેલ ગોળાના દ્રવ્ય માટે એકમ દળ દીઠ વિશિષ્ટ ઉષ્મા \(\alpha T^3\) મુજબ ફરે છે જ્યાં \(\alpha \) અચળાંક છે.તો ગોળાનું તાપમાન \(2T_0\) થતાં કેટલો સમય લાગશે?

A \(\frac{{M\alpha }}{{4\pi {R^2}\sigma }}\,\ln \left( {\frac{3}{2}} \right)\)
B \(\frac{{M\alpha }}{{4\pi {R^2}\sigma }}\,\ln \left( {\frac{16}{3}} \right)\)
C \(\frac{{M\alpha }}{{16\pi {R^2}\sigma }}\,\ln \left( {\frac{16}{3}} \right)\)
D \(\frac{{M\alpha }}{{16\pi {R^2}\sigma }}\,\ln \left( {\frac{3}{2}} \right)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{{M\alpha }}{{16\pi {R^2}\sigma }}\,\ln \left( {\frac{16}{3}} \right)\)

Step-by-step Solution

Detailed explanation

In the given problem, fall in temperature of sphere, \(dT = \left( {3{T_0} - 2{T_0}} \right) = {T_0}\) Tmperature of surrounding, \({T_{surr}} = {T_0}\) Initial temperature of sphere, \({T_{initial}} = 3{T_0}\) Specific heat of the material of the sphere varies as,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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