Diameter of two spheres of same material are in the ratio of 1:2 . If the temp of both sphere are same then the ratio of their heat energy will be

a) 1:2

b) 1:4

c) 1:8

d) 1:16

To determine the ratio of heat energy between two spheres made of the same material, we need to consider how heat energy is related to the volume of the spheres, since they are of different diameters. The key concept here is that the heat energy stored in an object is proportional to its mass, and mass is directly related to volume when the material is uniform.

Understanding Volume and Diameter

The volume \( V \) of a sphere is calculated using the formula:

\( V = \frac{4}{3} \pi r^3 \)

Where \( r \) is the radius of the sphere. Since the diameter is twice the radius, we can express the radius in terms of the diameter \( d \):

\( r = \frac{d}{2} \)

Thus, the volume can also be expressed in terms of the diameter:

\( V = \frac{1}{6} \pi d^3 \)

Calculating the Volumes

Given that the diameters of the two spheres are in the ratio of 1:2, we can denote the diameters as \( d_1 \) and \( d_2 \), where:

• Let \( d_1 = d \)
• Then \( d_2 = 2d \)

Now, we can find the volumes of both spheres:

• Volume of sphere 1: \( V_1 = \frac{1}{6} \pi d^3 \)
• Volume of sphere 2: \( V_2 = \frac{1}{6} \pi (2d)^3 = \frac{1}{6} \pi (8d^3) = \frac{4}{3} \pi d^3 \)

Finding the Ratio of Volumes

Now, let's find the ratio of their volumes:

\( \text{Ratio of volumes} = \frac{V_1}{V_2} = \frac{\frac{1}{6} \pi d^3}{\frac{4}{3} \pi d^3} = \frac{1}{8} \)

Relating Volume to Heat Energy

Since the spheres are made of the same material and are at the same temperature, the heat energy \( Q \) stored in each sphere is proportional to its volume:

\( Q \propto V \)

Thus, the ratio of their heat energies will also be the same as the ratio of their volumes:

\( \text{Ratio of heat energies} = \frac{Q_1}{Q_2} = \frac{V_1}{V_2} = \frac{1}{8} \)

Final Answer

Therefore, the ratio of the heat energy of the two spheres is:

1:8

So, the correct option is c) 1:8.