B

To calculate the ratio of energy radiated by two black bodies, we use the Stefan-Boltzmann Law, which states:

• The energy radiated per unit area of a black body is given by E = \sigma T^4, where \sigma is the Stefan-Boltzmann constant and T is the absolute temperature of the body in Kelvin.
• Assuming the spheres are black bodies, the total energy emitted by each sphere is given by Q = A E, where A is the surface area, which depends on the radius r as A = 4\pi r^2.

For Sphere 1 with radius \( r_1 \) and temperature \( T_1 \) and Sphere 2 with radius \( r_2 \) and temperature \( T_2 \):

• Radius of Sphere 1 = 2 cm = 0.02 m
• Radius of Sphere 2 = 8 cm = 0.08 m
• Temperature of Sphere 1 = 127 °C = 400 K
• Temperature of Sphere 2 = 527 °C = 800 K

Energy emitted by Sphere 1: $$Q_1 = (4\pi (0.02)^2) \sigma (400)^4$$ Energy emitted by Sphere 2: $$Q_2 = (4\pi (0.08)^2) \sigma (800)^4$$ Now, taking the ratio of the two: $$\frac{Q_1}{Q_2} = \frac{(0.02)^2 (400)^4}{(0.08)^2 (800)^4}$$ This simplifies to: $$\frac{Q_1}{Q_2} = \frac{(0.04) (6.4 \times 10^{10})}{(0.64) (4.096 \times 10^{10})} = \frac{1}{2}$$ Thus, the ratio of energy radiated by Sphere 1 to Sphere 2 is \(1:2\).
Therefore, the correct option is B.