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 analyze the dimensions of each component in the expression.

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

1. **Charge (e)**: The dimensional formula for electric charge is given by:
\[ [e] = [M^{0} L^{1} T^{-1} I^{1}] \]
where \(M\) is mass, \(L\) is length, \(T\) is time, and \(I\) is electric current.

2. **Permittivity of free space (\(\epsilon_0\))**: The dimensional formula for permittivity is:
\[ [\epsilon_0] = [M^{-1} L^{-3} T^{4} I^{2}] \]

3. **Universal gravitational constant (G)**: The dimensional formula for \(G\) is:
\[ [G] = [M^{-1} L^{3} T^{-2}] \]

4. **Mass of proton (\(m_p\))**: The dimensional formula for mass is:
\[ [m_p] = [M^{1}] \]

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

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

\[ [e^2] = [M^{0} L^{1} T^{-1} I^{1}]^2 = [M^{0} L^{2} T^{-2} I^{2}] \]

Now, substituting the dimensions into the expression:

\[ \frac{e^2}{4\pi \epsilon_0 G m_p^2} = \frac{[M^{0} L^{2} T^{-2} I^{2}]}{[M^{-1} L^{-3} T^{4} I^{2}] \cdot [M^{-1} L^{3} T^{-2}] \cdot [M^{2}]} \]

### Step 3: Combine the dimensions in the denominator

The denominator becomes:

\[ [\epsilon_0 G m_p^2] = [M^{-1} L^{-3} T^{4} I^{2}] \cdot [M^{-1} L^{3} T^{-2}] \cdot [M^{2}] \]

Calculating the dimensions step-by-step:

1. \(M^{-1} \cdot M^{-1} \cdot M^{2} = M^{0}\)
2. \(L^{-3} \cdot L^{3} = L^{0}\)
3. \(T^{4} \cdot T^{-2} = T^{2}\)
4. \(I^{2} = I^{2}\)

So, the overall dimensional formula for the denominator is:

\[ [M^{0} L^{0} T^{2} I^{2}] \]

### Step 4: Final calculation

Now we can write the overall dimensional formula:

\[ \frac{[M^{0} L^{2} T^{-2} I^{2}]}{[M^{0} L^{0} T^{2} I^{2}]} = [M^{0} L^{2} T^{-2} I^{2}] \cdot [M^{0} L^{0} T^{-2} I^{-2}] = [M^{0} L^{2} T^{-4} I^{0}] \]

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

\[ [L^{2} T^{-4}] \]

### Conclusion

The final answer is:
\[ \text{Dimensional formula: } [L^{2} T^{-4}] \]