Dimensional formula of `(e^(4))/(epsilon_(0)^(2)m_(p)m_(e)^(2)c^(3)G)` is (Where `e=` charge , `m_(p)` and `m_(c )` are masses, `epsilon_(0)=` permittivity of free space, `c=` velocity of light and `G=` gravitational constant)

Options

A. `M`
B. `T`
C. `p`
D. `I^(4)T^(2)M`

Answer

To find the dimensional formula of the expression \(\frac{e^4}{\epsilon_0^2 m_p m_e^2 c^3 G}\), we will break down each component into its respective dimensional formula and then combine them according to the expression.

### Step-by-Step Solution:

1. **Identify the Dimensional Formulas**:
- Charge \(e\): The dimensional formula for charge is given by \( [e] = [I][T^1] \) (where \(I\) is current and \(T\) is time).
- Permittivity of free space \(\epsilon_0\): The dimensional formula is \( [\epsilon_0] = [M^{-1} L^{-3} T^4 I^2] \).
- Mass \(m_p\) and \(m_e\): The dimensional formula for mass is \( [M] \).
- Velocity of light \(c\): The dimensional formula is \( [c] = [L T^{-1}] \).
- Gravitational constant \(G\): The dimensional formula is \( [G] = [M^{-1} L^3 T^{-2}] \).

2. **Substituting the Dimensional Formulas**:
- The expression can be rewritten as:
\[ \frac{[e]^4}{[\epsilon_0]^2 [m_p] [m_e]^2 [c]^3 [G]} \]

3. **Calculating Each Component**:
- For \(e^4\):
\[ [e^4] = [I^4][T^4] \]
- For \(\epsilon_0^2\):
\[ [\epsilon_0^2] = [M^{-2} L^{-6} T^8 I^4] \]
- For \(m_p\) and \(m_e^2\):
\[ [m_p] = [M], \quad [m_e^2] = [M^2] \]
- For \(c^3\):
\[ [c^3] = [L^3 T^{-3}] \]
- For \(G\):
\[ [G] = [M^{-1} L^3 T^{-2}] \]

4. **Combining the Dimensional Formulas**:
- Now substituting these into the expression:
\[ \frac{[I^4][T^4]}{[M^{-2} L^{-6} T^8 I^4][M][M^2][L^3 T^{-3}][M^{-1} L^3 T^{-2}]} \]

5. **Simplifying the Denominator**:
- Combine the mass terms:
\[ [M^{-2}][M][M^2][M^{-1}] = [M^{0}] = [1] \]
- Combine the length terms:
\[ [L^{-6}][L^3][L^3] = [L^{0}] = [1] \]
- Combine the time terms:
\[ [T^8][T^{-3}][T^{-2}] = [T^{3}] \]
- The \(I\) terms:
\[ [I^4] = [I^4] \]

6. **Final Expression**:
- Therefore, the overall dimensional formula becomes:
\[ \frac{[I^4][T^4]}{[1][1][T^3][I^4]} = [T^{4-3}] = [T^1] \]

### Conclusion:
The dimensional formula of \(\frac{e^4}{\epsilon_0^2 m_p m_e^2 c^3 G}\) is \([T]\).