If the energy, `E = G^p h^q c^r,` where G is the universal gravitational constant, h is the Planck's constant and c is the velocity of light, then the values of p are q and r are, respectively

Options

A. `-1//2, 1//2 and 5//2`
B. `1//2, -1//2 and -5//2`
C. `-1//2, 1//2 and 3//2`
D. `1//2, -1//2 and -3//2`

Explanation

To solve the problem of finding the values of \( p \), \( q \), and \( r \) in the equation \( E = G^p h^q c^r \), we will follow these steps:

### Step 1: Write the dimensions of energy \( E \)
Energy can be expressed in terms of mass, length, and time. The dimensional formula for energy \( E \) is:
\[ [E] = M L^2 T^{-2} \]

### Step 2: Write the dimensions of the constants \( G \), \( h \), and \( c \)
1. **Universal Gravitational Constant \( G \)**:
The gravitational force is given by \( F = \frac{G m_1 m_2}{r^2} \). Rearranging gives:
\[ G = \frac{F r^2}{m_1 m_2} \]
The dimension of force \( F \) is \( M L T^{-2} \), and the dimensions of \( m_1 \) and \( m_2 \) are \( M \). Therefore, the dimensions of \( G \) are:
\[ [G] = \frac{M L T^{-2} \cdot L^2}{M^2} = M^{-1} L^3 T^{-2} \]

2. **Planck's Constant \( h \)**:
Planck's constant relates energy and frequency: \( E = h \nu \), where \( \nu \) (frequency) has dimensions \( T^{-1} \). Thus:
\[ [h] = \frac{E}{\nu} = \frac{M L^2 T^{-2}}{T^{-1}} = M L^2 T^{-1} \]

3. **Speed of Light \( c \)**:
The speed of light has dimensions of length per time:
\[ [c] = L T^{-1} \]

### Step 3: Substitute the dimensions into the equation
Now we substitute the dimensions of \( G \), \( h \), and \( c \) into the equation \( E = G^p h^q c^r \):
\[ [M L^2 T^{-2}] = [G^p][h^q][c^r] \]
Substituting the dimensions we found:
\[ M L^2 T^{-2} = (M^{-1} L^3 T^{-2})^p (M L^2 T^{-1})^q (L T^{-1})^r \]
This expands to:
\[ M L^2 T^{-2} = M^{-p + q} L^{3p + 2q + r} T^{-2p - q - r} \]

### Step 4: Set up equations for each dimension
Now we equate the powers of \( M \), \( L \), and \( T \):
1. For \( M \):
\[ -p + q = 1 \quad \text{(Equation 1)} \]
2. For \( L \):
\[ 3p + 2q + r = 2 \quad \text{(Equation 2)} \]
3. For \( T \):
\[ -2p - q - r = -2 \quad \text{(Equation 3)} \]

### Step 5: Solve the equations
From Equation 1:
\[ q = p + 1 \]
Substituting \( q \) into Equation 2:
\[ 3p + 2(p + 1) + r = 2 \implies 3p + 2p + 2 + r = 2 \implies 5p + r = 0 \implies r = -5p \]
Substituting \( q \) and \( r \) into Equation 3:
\[ -2p - (p + 1) - (-5p) = -2 \implies -2p - p - 1 + 5p = -2 \implies 2p - 1 = -2 \implies 2p = -1 \implies p = -\frac{1}{2} \]
Now substituting \( p \) back to find \( q \) and \( r \):
\[ q = -\frac{1}{2} + 1 = \frac{1}{2} \]
\[ r = -5(-\frac{1}{2}) = \frac{5}{2} \]

### Final Values
Thus, the values of \( p \), \( q \), and \( r \) are:
\[ p = -\frac{1}{2}, \quad q = \frac{1}{2}, \quad r = \frac{5}{2} \]