In case of a dipole field

Options

A. intensity can be zero
B. potential can be zero
C. both can be zero
D. none of these

Answer

To solve the problem of finding the speed of each proton when they reach infinity due to their mutual repulsion, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the System**: We have two protons, each with charge \( q \) (where \( q = e = 1.6 \times 10^{-19} \, \text{C} \)), separated by a distance \( r \). They repel each other due to their like charges.

2. **Calculate the Electrostatic Force**: The electrostatic force \( F \) between the two protons can be calculated using Coulomb's law:
\[ F = \frac{k \cdot q_1 \cdot q_2}{r^2} = \frac{k \cdot q^2}{r^2} \]
where \( k \) is Coulomb's constant, \( k = \frac{1}{4 \pi \epsilon_0} \).

3. **Calculate the Initial Potential Energy**: The potential energy \( U \) of the system when the protons are separated by a distance \( r \) is given by:
\[ U = \frac{k \cdot q^2}{r} \]

4. **Energy Conservation Principle**: As the protons move apart due to their mutual repulsion, the potential energy converts into kinetic energy. At infinity, the potential energy approaches zero, and all the initial potential energy becomes kinetic energy.

5. **Kinetic Energy of Each Proton**: Since both protons will have the same kinetic energy \( K \) when they reach infinity, we can express this as:
\[ K = \frac{1}{2} m v^2 \]
where \( m \) is the mass of a proton and \( v \) is the speed of each proton.

6. **Set Initial Potential Energy Equal to Total Kinetic Energy**: At the initial position, all potential energy will convert into kinetic energy:
\[ \frac{k \cdot q^2}{r} = 2 \cdot \frac{1}{2} m v^2 \]
Simplifying gives:
\[ \frac{k \cdot q^2}{r} = m v^2 \]

7. **Solve for Speed \( v \)**: Rearranging the equation to solve for \( v \):
\[ v^2 = \frac{k \cdot q^2}{m \cdot r} \]
\[ v = \sqrt{\frac{k \cdot q^2}{m \cdot r}} \]

8. **Substituting \( k \)**: Substitute \( k = \frac{1}{4 \pi \epsilon_0} \):
\[ v = \sqrt{\frac{\frac{1}{4 \pi \epsilon_0} \cdot q^2}{m \cdot r}} \]

9. **Final Expression**: Therefore, the speed of each proton when they reach infinity is:
\[ v = \sqrt{\frac{q^2}{4 \pi \epsilon_0 m r}} \]