Two spheres A and B have diameters in the ratio `1:2`, densities in the ratio `2:1` and specific heat in the ratio `1:3`. Find the ratio of their thermal capacities.

Options

A. `1:6`
B. `1:12`
C. `1:3`
D. `1:4`

Answer

To find the ratio of thermal capacities of two spheres A and B, we will follow these steps:

### Step 1: Understand the formula for thermal capacity
The thermal capacity (C) of an object is given by the formula:
\[ C = m \cdot S \]
where \( m \) is the mass of the object and \( S \) is the specific heat capacity.

### Step 2: Calculate the mass of the spheres
The mass \( m \) of a sphere can be calculated using the formula:
\[ m = \rho \cdot V \]
where \( \rho \) is the density and \( V \) is the volume of the sphere. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Given that the diameter ratio of spheres A and B is \( 1:2 \), we can express the radii as:
- \( d_A = d \) (for sphere A)
- \( d_B = 2d \) (for sphere B)

Thus, the radii are:
- \( r_A = \frac{d}{2} \)
- \( r_B = d \)

### Step 3: Calculate the volume of the spheres
Now, substituting the radius into the volume formula:
- For sphere A:
\[ V_A = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 = \frac{4}{3} \pi \frac{d^3}{8} = \frac{1}{6} \pi d^3 \]
- For sphere B:
\[ V_B = \frac{4}{3} \pi (d)^3 = \frac{4}{3} \pi d^3 \]

### Step 4: Calculate the mass of the spheres using their densities
Given the density ratio \( \rho_A : \rho_B = 2:1 \), we can express:
- \( \rho_A = 2 \rho \)
- \( \rho_B = \rho \)

Now, substituting the densities and volumes into the mass formula:
- For sphere A:
\[ m_A = \rho_A \cdot V_A = (2\rho) \cdot \left(\frac{1}{6} \pi d^3\right) = \frac{1}{3} \pi \rho d^3 \]
- For sphere B:
\[ m_B = \rho_B \cdot V_B = \rho \cdot \left(\frac{4}{3} \pi d^3\right) = \frac{4}{3} \pi \rho d^3 \]

### Step 5: Calculate the thermal capacities of the spheres
Now substituting the masses and specific heat capacities into the thermal capacity formula. Given the specific heat ratio \( S_A : S_B = 1:3 \):
- \( S_A = S \)
- \( S_B = 3S \)

Thus, the thermal capacities are:
- For sphere A:
\[ C_A = m_A \cdot S_A = \left(\frac{1}{3} \pi \rho d^3\right) \cdot S = \frac{1}{3} \pi \rho d^3 S \]
- For sphere B:
\[ C_B = m_B \cdot S_B = \left(\frac{4}{3} \pi \rho d^3\right) \cdot (3S) = 4 \pi \rho d^3 S \]

### Step 6: Find the ratio of thermal capacities
Now, we can find the ratio of thermal capacities:
\[ \frac{C_A}{C_B} = \frac{\frac{1}{3} \pi \rho d^3 S}{4 \pi \rho d^3 S} \]

The \( \pi \), \( \rho \), and \( d^3 S \) terms cancel out:
\[ \frac{C_A}{C_B} = \frac{1/3}{4} = \frac{1}{12} \]

### Final Result
Thus, the ratio of the thermal capacities of spheres A and B is:
\[ C_A : C_B = 1 : 12 \]

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