The time dependence of a physical quantity P is given by `P=P_0 exp(prop t^2)`, where `prop` is constant `prop` is represented as `[M^0 L^x T^(-2)]`. Find x

Answer

To solve the problem, we need to analyze the given expression for the physical quantity \( P \):

\[ P = P_0 \exp(\text{prop} \cdot t^2) \]

where \( \text{prop} \) is a constant with dimensions represented as \( [M^0 L^x T^{-2}] \). Our goal is to find the value of \( x \).

### Step-by-step Solution:

1. **Understanding the Exponential Function**:
The argument of the exponential function must be dimensionless. Therefore, the term \( \text{prop} \cdot t^2 \) must be dimensionless.

2. **Identifying the Dimensions**:
- The dimension of time \( t \) is represented as \( [T] \).
- Consequently, the dimension of \( t^2 \) is \( [T^2] \).

3. **Setting Up the Equation**:
Since \( \text{prop} \cdot t^2 \) is dimensionless, we can write:
\[ [\text{prop}] \cdot [t^2] = [M^0 L^x T^{-2}] \cdot [T^2] = [M^0 L^x T^{0}] \]
This means that the dimensions of \( \text{prop} \) multiplied by the dimensions of \( t^2 \) must equal \( [M^0 L^0 T^0] \).

4. **Equating Dimensions**:
From the above, we have:
\[ [M^0 L^x T^{0}] = [M^0 L^0 T^0] \]
This implies:
- For mass \( M \): \( 0 = 0 \) (holds true)
- For length \( L \): \( x = 0 \)
- For time \( T \): \( 0 = 0 \) (holds true)

5. **Conclusion**:
Thus, we find that \( x = 0 \).

### Final Answer:
The value of \( x \) is \( 0 \).

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