Heat and Thermal capacity differ in the dimensions of

Options

A. Mass
B. length
C. Time
D. Temperature

Answer

To solve the question regarding the difference in dimensions between heat and thermal capacity, we will follow these steps:

### Step 1: Understand the Definitions
- **Heat** is a form of energy that is transferred between systems or objects with different temperatures.
- **Thermal Capacity** (or heat capacity) is the amount of heat required to change the temperature of an object by one degree Celsius (or one Kelvin).

### Step 2: Write the Dimensional Formula for Heat
The dimensional formula for heat can be derived from the formula for energy. The formula for energy is given as:
\[ \text{Energy} = \text{Force} \times \text{Distance} \]
Where:
- Force = mass × acceleration = \( m \cdot a \)
- Acceleration = \( \frac{\text{change in velocity}}{\text{time}^2} \)

Thus, the dimensional formula for heat (energy) is:
\[ [\text{Heat}] = [\text{Energy}] = [M^1 L^2 T^{-2}] \]

### Step 3: Write the Dimensional Formula for Thermal Capacity
Thermal capacity is defined as the amount of heat required to change the temperature of a substance. The formula for thermal capacity is:
\[ \text{Thermal Capacity} = \frac{\text{Heat}}{\Delta T} \]
Where \( \Delta T \) is the change in temperature.

Using the dimensional formula for heat:
\[ [\text{Thermal Capacity}] = \frac{[M^1 L^2 T^{-2}]}{[\Theta]} \]
Where \( [\Theta] \) represents the dimension of temperature.

Thus, the dimensional formula for thermal capacity becomes:
\[ [\text{Thermal Capacity}] = [M^1 L^2 T^{-2} \Theta^{-1}] \]

### Step 4: Compare the Dimensional Formulas
Now we can compare the two dimensional formulas:
- Dimensional formula for heat: \( [M^1 L^2 T^{-2}] \)
- Dimensional formula for thermal capacity: \( [M^1 L^2 T^{-2} \Theta^{-1}] \)

### Step 5: Identify the Differences
From the comparison, we can see that:
- The dimensions of mass (M), length (L), and time (T) are the same in both formulas.
- The only difference is the presence of \( \Theta^{-1} \) in the dimensional formula for thermal capacity, indicating that thermal capacity has a dimension related to temperature.

### Conclusion
Therefore, heat and thermal capacity differ in the dimensions of temperature.

### Final Answer
Heat and thermal capacity differ in the dimensions of **temperature** (Option 4).

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