If a proton in an elctorn an an elctron are accelerated through the same potential difference

Options

A. both the proton and electron have same K.E
B. both the proton and electron have same momentum
C. both the proton and electron have same velocity
D. both the proton and electron have same temperature

Answer

To solve the problem of whether a proton and an electron accelerated through the same potential difference have the same kinetic energy, momentum, velocity, or temperature, we can analyze each aspect step by step.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
- When a charged particle is accelerated through a potential difference \( V \), the kinetic energy (\( KE \)) gained by the particle is given by the equation:
\[ KE = Q \cdot V \]
- Here, \( Q \) is the charge of the particle.

2. **Calculating Kinetic Energy for Proton and Electron**:
- The charge of a proton (\( Q_p \)) is \( +1.6 \times 10^{-19} \) C.
- The charge of an electron (\( Q_e \)) is \( -1.6 \times 10^{-19} \) C (magnitude is the same).
- Therefore, the kinetic energy for both the proton and the electron when accelerated through the same potential difference \( V \) is:
\[ KE_p = Q_p \cdot V = (1.6 \times 10^{-19}) \cdot V \]
\[ KE_e = Q_e \cdot V = (1.6 \times 10^{-19}) \cdot V \]
- Since both expressions are equal, we conclude:
\[ KE_p = KE_e \]
- **Conclusion**: Both the proton and electron have the same kinetic energy.

3. **Analyzing Momentum**:
- Momentum (\( p \)) is given by:
\[ p = m \cdot v \]
- The kinetic energy can also be expressed in terms of momentum:
\[ KE = \frac{p^2}{2m} \]
- Since the kinetic energies are the same, we can express momentum for both particles:
\[ p_p = \sqrt{2 \cdot m_p \cdot KE_p} \]
\[ p_e = \sqrt{2 \cdot m_e \cdot KE_e} \]
- Given that the masses of the proton (\( m_p \)) and electron (\( m_e \)) are different (\( m_p \) is much larger than \( m_e \)), the momenta cannot be the same. Thus:
- **Conclusion**: The proton and electron do not have the same momentum.

4. **Analyzing Velocity**:
- The velocity can be derived from kinetic energy:
\[ KE = \frac{1}{2} m v^2 \implies v = \sqrt{\frac{2 \cdot KE}{m}} \]
- Since both particles have the same kinetic energy but different masses, their velocities will also differ:
\[ v_p = \sqrt{\frac{2 \cdot KE_p}{m_p}} \quad \text{and} \quad v_e = \sqrt{\frac{2 \cdot KE_e}{m_e}} \]
- **Conclusion**: The proton and electron do not have the same velocity.

5. **Analyzing Temperature**:
- Temperature is a measure of the average kinetic energy of particles in a system, but it is not directly related to the kinetic energy of individual particles in this context.
- Therefore, we cannot conclude that both particles have the same temperature based on the information given.
- **Conclusion**: The proton and electron do not have the same temperature.

### Final Conclusion:
- The correct option is that both the proton and electron have the same kinetic energy.