The ratio of electrostatic and gravitational force acting between electron and proton separated by a distance `5 xx 10^(-11)m`, will be (charge on electron `= 1.6 xx 10^(-19)C`, mass of electron `= 9.1 xx 10^(-31) kg`, mass of proton `= 1.6 xx 10^(-27) kg, G = 6.7 xx 10^(-11) N - m^(2)//kg^(2)`)

Options

A. `2.36 xx 10^(39)`
B. `2.36 xx 10^(40)`
C. `2.34 xx 10^(41)`
D. `2.34 xx 10^(42)`

Answer

To find the ratio of the electrostatic force to the gravitational force acting between an electron and a proton separated by a distance of \(5 \times 10^{-11} \, \text{m}\), we will use the formulas for both forces.

### Step 1: Calculate the Electrostatic Force
The electrostatic force \(F_e\) between two charges is given by Coulomb's law:
\[ F_e = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]
where:
- \(k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\) (Coulomb's constant),
- \(q_1 = -1.6 \times 10^{-19} \, \text{C}\) (charge of electron),
- \(q_2 = +1.6 \times 10^{-19} \, \text{C}\) (charge of proton),
- \(r = 5 \times 10^{-11} \, \text{m}\) (distance between them).

Substituting the values:
\[ F_e = \frac{(8.99 \times 10^9) \cdot (1.6 \times 10^{-19}) \cdot (1.6 \times 10^{-19})}{(5 \times 10^{-11})^2} \]

Calculating the denominator:
\[ (5 \times 10^{-11})^2 = 25 \times 10^{-22} = 2.5 \times 10^{-21} \]

Now substituting back:
\[ F_e = \frac{(8.99 \times 10^9) \cdot (2.56 \times 10^{-38})}{2.5 \times 10^{-21}} \]
\[ F_e = \frac{(8.99 \times 2.56) \times 10^{-29}}{2.5 \times 10^{-21}} = \frac{23.0544 \times 10^{-29}}{2.5 \times 10^{-21}} = 9.22176 \times 10^{-8} \, \text{N} \]

### Step 2: Calculate the Gravitational Force
The gravitational force \(F_g\) between two masses is given by Newton's law of gravitation:
\[ F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \(G = 6.7 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\),
- \(m_1 = 9.1 \times 10^{-31} \, \text{kg}\) (mass of electron),
- \(m_2 = 1.6 \times 10^{-27} \, \text{kg}\) (mass of proton),
- \(r = 5 \times 10^{-11} \, \text{m}\).

Substituting the values:
\[ F_g = \frac{(6.7 \times 10^{-11}) \cdot (9.1 \times 10^{-31}) \cdot (1.6 \times 10^{-27})}{(5 \times 10^{-11})^2} \]
Using the previously calculated denominator:
\[ F_g = \frac{(6.7 \times 10^{-11}) \cdot (1.456 \times 10^{-57})}{2.5 \times 10^{-21}} = \frac{9.76 \times 10^{-68}}{2.5 \times 10^{-21}} = 3.904 \times 10^{-47} \, \text{N} \]

### Step 3: Calculate the Ratio of Electrostatic to Gravitational Force
Now we can find the ratio \(R\):
\[ R = \frac{F_e}{F_g} = \frac{9.22176 \times 10^{-8}}{3.904 \times 10^{-47}} \]
Calculating the ratio:
\[ R = 2.36 \times 10^{39} \]

### Final Answer
The ratio of the electrostatic force to the gravitational force acting between the electron and proton is:
\[ \boxed{2.36 \times 10^{39}} \]