If a and b are two physical quantities having different dimensions then which of the following can denote a new physical quantity

Options

A. `a+b`
B. `a-b`
C. `a//b`
D. `e^(a//b)`

Answer

To determine which of the given options can denote a new physical quantity when two physical quantities \( a \) and \( b \) have different dimensions, we can analyze the possible operations that can be performed on these quantities.

### Step-by-Step Solution:

1. **Identify the Physical Quantities**:
Let \( a \) and \( b \) be two physical quantities with different dimensions. For example, let \( a \) be a length (with dimension \( [L] \)) and \( b \) be a time (with dimension \( [T] \)).

2. **Consider Addition and Subtraction**:
- When we try to add or subtract two physical quantities with different dimensions (e.g., \( a + b \) or \( a - b \)), it is not meaningful because the dimensions do not match.
- Therefore, \( a + b \) or \( a - b \) cannot represent a new physical quantity.

3. **Consider Multiplication and Division**:
- If we divide \( a \) by \( b \) (i.e., \( \frac{a}{b} \)), we get a new quantity with dimensions \( \frac{[L]}{[T]} \), which is the dimension of velocity \( [V] \).
- Similarly, if we multiply \( a \) and \( b \) (i.e., \( a \cdot b \)), we would get a new quantity with dimensions \( [L][T] \), which could represent a different physical quantity depending on the context.

4. **Exponential Functions**:
- If we consider an expression like \( e^{\frac{a}{b}} \), where \( \frac{a}{b} \) has dimensions, this does not yield a meaningful physical quantity because the exponent in an exponential function must be dimensionless. Hence, \( e^{\frac{a}{b}} \) does not represent a new physical quantity.

5. **Conclusion**:
- The only operation that results in a meaningful new physical quantity when combining two physical quantities of different dimensions is division. Therefore, the correct option is the one that represents a division of the two quantities.

### Final Answer:
The operation that can denote a new physical quantity when \( a \) and \( b \) have different dimensions is \( \frac{a}{b} \).