If the radii of two copper spheres are in the ratio 1:3 and their increase in the temperatures are in the ratio 9:1 then the ratio of the increase in their thermal energy will be

a.1:4

b.2:1

c.4:1

d.1:3

To find the ratio of the increase in thermal energy of the two copper spheres, we need to consider a few key concepts related to thermal energy, specifically how it relates to mass, temperature change, and specific heat capacity. Let's break this down step by step.

Understanding Thermal Energy

The thermal energy (Q) of an object can be calculated using the formula:

Q = mcΔT

• m = mass of the object
• c = specific heat capacity (which is constant for a given material)
• ΔT = change in temperature

Finding the Mass of Each Sphere

The mass of a sphere can be determined using the formula for the volume of a sphere and the density of the material:

Volume (V) = (4/3)πr³

Since the density of copper is constant, the mass will be proportional to the volume. Given that the radii of the spheres are in the ratio of 1:3, we can express their volumes and thus their masses in terms of the radius.

Let the radius of the first sphere be r and the radius of the second sphere be 3r. The volumes will then be:

• Volume of first sphere: V₁ = (4/3)πr³
• Volume of second sphere: V₂ = (4/3)π(3r)³ = 36(4/3)πr³

This means the mass of the first sphere (m₁) is proportional to r³ and the mass of the second sphere (m₂) is proportional to 36r³. Therefore, the ratio of their masses is:

m₁:m₂ = 1:36

Considering Temperature Changes

The problem states that the increase in temperatures of the two spheres is in the ratio of 9:1. Therefore, we can express the temperature changes as:

• ΔT₁ = 9x
• ΔT₂ = x

Calculating the Increase in Thermal Energy

Now we can calculate the thermal energy increase for each sphere:

• For the first sphere: Q₁ = m₁cΔT₁ = (1)(c)(9x) = 9cx
• For the second sphere: Q₂ = m₂cΔT₂ = (36)(c)(x) = 36cx

Finding the Ratio of Thermal Energies

Now, we can find the ratio of the increases in thermal energy:

Q₁:Q₂ = 9cx:36cx = 9:36 = 1:4

Final Answer

The ratio of the increase in their thermal energy is 1:4, which corresponds to option a.