Question

A spherical black body with a radius of 12 cm radiates 450 W power at 50 K. If the radius were halved and the temperature doubled, the power radiated in watts would be
A. 225
B. 450
C. 900
D. 1800

Answer 1

Hint: First of all, find the new radius and temperature by the given conditions. Then use the formula of power radiated for a black body (in watts) to obtain the required answer by the given data.

Formula used: \[P = A\sigma {T^4}\],\[A = 4\pi {R^2}\] and \[P = 4\pi {R^2}\sigma {T^4}\]

Complete Step-by-Step solution:
Given that
Radius of spherical black body \[R = 12\]
Given the new radius were halved i.e., \[r = \dfrac{R}{2} = \dfrac{{12}}{2} = 6\]
Temperature of the spherical black body \[T = 50\]
Given new temperature of the black body is doubled i.e., \[t = 2 \times T = 2 \times 50 = 100\]
Power radiated by the spherical black body \[P = 450\]
We know that the power radiated by the black body is given by \[P = A\sigma {T^4}\] where \[P\] is the net radiated power, \[A\] is radiating area and \[T\] is the temperature of radiator.
Also, we know that for a spherical black body \[A = 4\pi {R^2}\] where \[R\] is the radius of spherical black body.
So, we have the formula for power radiated by the black \[P = 4\pi {R^2}\sigma {T^4}\]
And given that \[P = 4\pi {R^2}\sigma {T^4} = 450........................................\left( 1 \right)\]
As we have \[r = \dfrac{R}{2}{\text{ and }}t = 2T\] the power radiated by the new spherical black body is given by
\[
  {P_{{\text{new}}}} = 4\pi {r^2}\sigma {t^4} \\
  {P_{{\text{new}}}} = 4\pi {\left( {\dfrac{R}{2}} \right)^2}\sigma {\left( {2T} \right)^4} \\
  {P_{{\text{new}}}} = 4\pi {R^2}\sigma {T^4}\left( {\dfrac{{{2^4}}}{{{2^2}}}} \right) \\
  {P_{{\text{new}}}} = 450\left( {{2^2}} \right){\text{ }}\left[ {{\text{Using }}\left( 1 \right)} \right] \\
  {P_{{\text{new}}}} = 450\left( 4 \right) \\
  \therefore {P_{{\text{new}}}} = 1800{\text{ watts}} \\
\]
Hence the power radiated by the new spherical black body is 1800 watts
Thus, the correct option is D. 1800

Note: A black body is defined as one that absorbs all incident radiation so that all the radiation that comes from its surface is its own emission. The name ‘black body’ is given because it absorbs radiation in all frequencies, not because it only absorbs. Indeed, a black body can also emit radiation.