If `epsilon_(0)` is permittivity of free space, e is charge of proton, G is universal gravitational constant and `m_(p)` is mass of a proton then the dimensional formula for
`(e^(2))/(4pi epsilon_(0)Gm_(p)^(2))` is

Options

A. `[M^(1)L^(1)T^(-3)A^(-1)]`
B. `[M^(0)L^(0)T^(0)A^(0)]`
C. `[M^(1)L^(3)T^(-3)A^(-1)]`
D. `[M^(-1)L^(-3)T^(4)A^(2)]`

Answer

To find the dimensional formula for the expression \(\frac{e^2}{4\pi \epsilon_0 G m_p^2}\), we will break down each component of the expression step by step.

### Step 1: Identify the dimensions of each quantity

1. **Charge of Proton (\(e\))**:
- The dimension of electric charge is given as \([A T]\), where \(A\) is the dimension of electric current and \(T\) is the dimension of time.

2. **Permittivity of Free Space (\(\epsilon_0\))**:
- From Coulomb's law, we have:
\[ F = \frac{1}{4\pi \epsilon_0} \frac{Q_1 Q_2}{r^2} \]
- Rearranging gives:
\[ \epsilon_0 = \frac{Q_1 Q_2}{F r^2} \]
- The dimensions of force \(F\) are \([M L T^{-2}]\), and distance \(r\) has dimensions \([L]\). The charge dimensions are \([A T]\).
- Thus, substituting in gives:
\[ \epsilon_0 = \frac{(A T)(A T)}{(M L T^{-2})(L^2)} = \frac{A^2 T^2}{M L^3 T^{-2}} = \frac{A^2 T^4}{M L^3} \]
- Therefore, the dimension of \(\epsilon_0\) is:
\[ [\epsilon_0] = [M^{-1} L^{-3} T^4 A^2] \]

3. **Universal Gravitational Constant (\(G\))**:
- From the formula for gravitational force:
\[ F = \frac{G m_1 m_2}{r^2} \]
- Rearranging gives:
\[ G = \frac{F r^2}{m_1 m_2} \]
- Substituting the dimensions:
\[ G = \frac{(M L T^{-2}) L^2}{M^2} = \frac{M L^3 T^{-2}}{M^2} = [M^{-1} L^3 T^{-2}] \]

4. **Mass of Proton (\(m_p\))**:
- The dimension of mass is simply:
\[ [m_p] = [M] \]

### Step 2: Substitute the dimensions into the expression

Now we substitute the dimensions into the expression \(\frac{e^2}{4\pi \epsilon_0 G m_p^2}\):

\[ \frac{e^2}{4\pi \epsilon_0 G m_p^2} = \frac{(A T)^2}{\epsilon_0 G (M^2)} \]

### Step 3: Write the dimensions of the entire expression

Substituting the dimensions we found:

1. For \(e^2\):
\[ [e^2] = [A^2 T^2] \]

2. For \(\epsilon_0\):
\[ [\epsilon_0] = [M^{-1} L^{-3} T^4 A^2] \]

3. For \(G\):
\[ [G] = [M^{-1} L^3 T^{-2}] \]

4. For \(m_p^2\):
\[ [m_p^2] = [M^2] \]

### Step 4: Combine the dimensions

Now we can combine these dimensions in the expression:

\[ \text{Denominator: } [\epsilon_0 G m_p^2] = [M^{-1} L^{-3} T^4 A^2] \cdot [M^{-1} L^3 T^{-2}] \cdot [M^2] \]

Calculating the denominator:

\[ = [M^{-1} L^{-3} T^4 A^2] \cdot [M^{-1} L^3 T^{-2}] \cdot [M^2] \]
\[ = [M^{-1} L^{-3} T^4 A^2] \cdot [M^{0} L^{0} T^{2}] = [M^{0} L^{0} T^{6} A^{2}] \]

### Step 5: Final expression

Now we can write the full expression:

\[ \frac{[A^2 T^2]}{[M^{0} L^{0} T^{6} A^{2}]} = [M^{0} L^{0} T^{-4}] \]

### Conclusion

Thus, the dimensional formula for \(\frac{e^2}{4\pi \epsilon_0 G m_p^2}\) is:

\[ [M^0 L^0 T^0] \quad \text{(dimensionless)} \]