If P represents radiation pressure, C speed of light, and Q radiation energy striking unit area per second and x,y,z are non zero integers, then P^x Q^y C^z is dimensionless. The values of X,y and z are respectively

Options

A. 1,1,-1
B. 1,-1,1
C. -1,1,1
D. 1,1,1

Answer

To solve the problem, we need to find the values of \( x \), \( y \), and \( z \) such that the expression \( P^x Q^y C^z \) is dimensionless. We will start by determining the dimensions of the quantities involved.

### Step 1: Determine the dimensions of each quantity.

1. **Radiation Pressure (P)**:
- The dimension of pressure is given by:
\[ [P] = \frac{[F]}{[A]} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

2. **Radiation Energy per unit area per unit time (Q)**:
- The dimension of energy is \( [M][L^2][T^{-2}] \).
- Therefore, the dimension of \( Q \) is:
\[ [Q] = \frac{[M][L^2][T^{-2}]}{[L^2][T]} = [M][T^{-3}] \]

3. **Speed of Light (C)**:
- The dimension of speed is:
\[ [C] = [L][T^{-1}] \]

### Step 2: Write the expression for \( P^x Q^y C^z \).

Now we can express the dimensions of \( P^x Q^y C^z \):
\[ [P^x] = ([M][L^{-1}][T^{-2}])^x = [M^x][L^{-x}][T^{-2x}] \]
\[ [Q^y] = ([M][T^{-3}])^y = [M^y][T^{-3y}] \]
\[ [C^z] = ([L][T^{-1}])^z = [L^z][T^{-z}] \]

Combining these, we have:
\[ [P^x Q^y C^z] = [M^{x+y}][L^{-x+z}][T^{-2x-3y-z}] \]

### Step 3: Set the dimensions equal to zero for a dimensionless quantity.

For the expression to be dimensionless, the exponents of \( M \), \( L \), and \( T \) must all equal zero:
1. \( x + y = 0 \) (1)
2. \( -x + z = 0 \) (2)
3. \( -2x - 3y - z = 0 \) (3)

### Step 4: Solve the equations.

From equation (1):
\[ y = -x \]

Substituting \( y = -x \) into equation (2):
\[ -x + z = 0 \implies z = x \]

Now substituting \( y = -x \) and \( z = x \) into equation (3):
\[ -2x - 3(-x) - x = 0 \implies -2x + 3x - x = 0 \implies 0 = 0 \]
This equation is satisfied for any \( x \).

### Step 5: Choose a value for \( x \).

Let’s choose \( x = 1 \):
- From \( y = -x \), we get \( y = -1 \).
- From \( z = x \), we get \( z = 1 \).

Thus, the values of \( x \), \( y \), and \( z \) are:
\[ x = 1, \quad y = -1, \quad z = 1 \]

### Final Answer:
The values of \( x \), \( y \), and \( z \) are respectively:
\[ \boxed{1, -1, 1} \]