The radiation emitted per unit time by unit area of a black body at temperature T is `sigmaT^(4)` wehre `sigma` is the Stefan-Boltzmann constant. The constant `sigma` can also be expressed in terms of Boltmann's constant (k), Planck's constant (h) and speed of light (3) as `sigma=Ak^(alpha)h^(beta)c^(gamma)` where A,`alpha,beta` and `gamma` are dimensionless constants. The set `(alpha,beta,gamma)` is givne by

Options

A. The electric field between the plates points vertically upward.
B. The pellet is negatively charged.
C. The magnitude of charge on the plates is increased, the pellet begins to move upward.
D. If the magnitude of charge on the plates is increased, the pellet begins to move upward.

Explanation

To find the values of \( \alpha \), \( \beta \), and \( \gamma \) in the expression for the Stefan-Boltzmann constant \( \sigma \), we start by equating the dimensions on both sides of the equation:

\[ \sigma = A k^{\alpha} h^{\beta} c^{\gamma} \]

### Step 1: Determine the dimensions of each constant

1. **Stefan-Boltzmann constant \( \sigma \)**:
- The dimension of \( \sigma \) is given as \( [\sigma] = ML^0T^{-3} \Theta^{-4} \).

2. **Boltzmann constant \( k \)**:
- The dimension of \( k \) is \( [k] = ML^2T^{-2}\Theta^{-1} \).

3. **Planck's constant \( h \)**:
- The dimension of \( h \) is \( [h] = ML^2T^{-1} \).

4. **Speed of light \( c \)**:
- The dimension of \( c \) is \( [c] = LT^{-1} \).

### Step 2: Write the dimensions of the right-hand side

Now, substituting the dimensions into the right-hand side:

\[ [A k^{\alpha} h^{\beta} c^{\gamma}] = [A] \cdot [k]^{\alpha} \cdot [h]^{\beta} \cdot [c]^{\gamma} \]

Assuming \( A \) is dimensionless, we have:

\[ = (ML^2T^{-2}\Theta^{-1})^{\alpha} \cdot (ML^2T^{-1})^{\beta} \cdot (LT^{-1})^{\gamma} \]

### Step 3: Expand the dimensions

Expanding this gives:

\[ = M^{\alpha + \beta} L^{2\alpha + 2\beta + \gamma} T^{-2\alpha - \beta - \gamma} \Theta^{-\alpha} \]

### Step 4: Set up equations by equating dimensions

Now we equate the dimensions from both sides:

1. For mass \( M \):
\[ \alpha + \beta = 1 \quad \text{(1)} \]

2. For length \( L \):
\[ 2\alpha + 2\beta + \gamma = 0 \quad \text{(2)} \]

3. For time \( T \):
\[ -2\alpha - \beta - \gamma = -3 \quad \text{(3)} \]

4. For temperature \( \Theta \):
\[ -\alpha = -4 \quad \text{(4)} \]

### Step 5: Solve the equations

From equation (4):
\[ \alpha = 4 \]

Substituting \( \alpha = 4 \) into equation (1):
\[ 4 + \beta = 1 \implies \beta = 1 - 4 = -3 \]

Substituting \( \alpha = 4 \) and \( \beta = -3 \) into equation (2):
\[ 2(4) + 2(-3) + \gamma = 0 \implies 8 - 6 + \gamma = 0 \implies \gamma = -2 \]

### Final Result

Thus, the values of the constants are:
\[ \alpha = 4, \quad \beta = -3, \quad \gamma = -2 \]