Let P represent radiation pressure, c represent speed of light and l represents radiation energy striking a unit area per second, then `P^(x//y)c^(z)` will be dimensionless for

Options

A. x = 0, y = z
B. x = y = z
C. x = z = -y
D. x = y = -z

Answer

To solve the problem, we need to determine the values of \(x\), \(y\), and \(z\) such that the expression \(P^{(x/y)} c^{z}\) is dimensionless. We will use the dimensional formulas of radiation pressure \(P\), radiation energy per unit area per second \(I\) (intensity), and the speed of light \(c\).

### Step-by-Step Solution:

1. **Identify the Dimensional Formulas:**
- The dimensional formula for radiation pressure \(P\) is:
\[ [P] = M L^{-1} T^{-2} \]
- The dimensional formula for intensity \(I\) (radiation energy striking a unit area per second) is:
\[ [I] = M T^{-3} \]
- The dimensional formula for the speed of light \(c\) is:
\[ [c] = L T^{-1} \]

2. **Express the Dimensions of the Given Expression:**
We need to analyze the expression \(P^{(x/y)} c^{z}\):
\[ P^{(x/y)} = (M L^{-1} T^{-2})^{(x/y)} = M^{(x/y)} L^{-(x/y)} T^{-(2x/y)} \]
\[ c^{z} = (L T^{-1})^{z} = L^{z} T^{-z} \]

3. **Combine the Dimensions:**
Now, we combine the dimensions from both parts:
\[ P^{(x/y)} c^{z} = M^{(x/y)} L^{-(x/y) + z} T^{-(2x/y) - z} \]

4. **Set the Exponents to Zero:**
For the expression to be dimensionless, all the exponents must equal zero:
- For \(M\):
\[ \frac{x}{y} = 0 \quad \text{(1)} \]
- For \(L\):
\[ -\frac{x}{y} + z = 0 \quad \text{(2)} \]
- For \(T\):
\[ -\frac{2x}{y} - z = 0 \quad \text{(3)} \]

5. **Solve the Equations:**
From equation (1), we see that \(x = 0\).

Substituting \(x = 0\) into equations (2) and (3):
- From (2):
\[ z = 0 \]
- From (3):
\[ -z = 0 \quad \Rightarrow \quad z = 0 \]

Since \(x = 0\) and \(z = 0\), we can substitute these values into the equations to find \(y\):
- From (1):
\[ \frac{0}{y} = 0 \quad \Rightarrow \quad y \text{ can be any value.} \]

6. **Final Relationships:**
We have established that:
\[ x = z = 0, \quad y \text{ can be any value.} \]

### Conclusion:
The expression \(P^{(x/y)} c^{z}\) is dimensionless when \(x = z = 0\) and \(y\) can be any value.