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If the inner and outer walls of a sphere having surface areas of A1 and A2 and inner and outer radii r1 and r2 are maintained at t1 and t2, then rate of heat flow will be 
1. \(\frac{k}{\sqrt{A_1A_2}}\frac{t_1\ -\ t_2}{r_2\ -\ r_1}\)
2. \(k\sqrt{A_1A_2}\frac{t_1\ -\ t_2}{r_2\ -\ r_1}\)
3. \(4\pi k\frac{t_1\ -\ t_2}{\sqrt{A_1A_2}}\)
4. \(4\pi kr_1r_2\frac{t_1\ -\ t_2}{\sqrt{A_1A_2}}\)

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Correct Answer - Option 2 : \(k\sqrt{A_1A_2}\frac{t_1\ -\ t_2}{r_2\ -\ r_1}\)

Explanation:

Heat conduction: 

The transfer of heat between two different bodies or two different locations of the same body through molecular vibrations is known as heat conduction.

The governing law for heat conduction is Fourier's law of heat conduction.

Fourier's law of heat conduction:

It states that 'The rate of heat transfer is directly proportional to the temperature gradient' and it is given as 

\(q_k = -k\ × \frac{dT}{dx}\)

Where x = direction of heat flow, k = thermal conductivity, T = temperature

The heat conduction equation of the spherical wall is given as:

\(Q = \frac{4\pi k(t_1\;-\;t_2)r_1r_2}{(r_2\;-\;r_1)}\)

So if we consider the surface areas at radius r1 and r2 then A1 = 4πr21 and A2 = 4πr22

\(Q = k \times \sqrt{4π r^2_1 \times 4π r^2_2} \times \frac{t_1\ -\ t_2 }{r_2\ -\ r_1}\)

∴ The heat conduction in a sphere having surface areas of A1 and A2 and inner and outer radii r1 and r2 are maintained at temperatures t1 and t2 is given as:

\(Q = k\ \times \sqrt{A_1 \times A_2} \times \frac{t_1\ -\ t_2 }{r_2\ -\ r_1}\)

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