An electron and a proton starting from rest are accelerated through a potential difference of 1000 V. Which one of the following statements in this regard is correct?
The speed of the electron will behigher than that of the proton.
Analyzing Electron and Proton Acceleration
This question asks us to compare the kinetic energy and speed of an electron and a proton when they are accelerated from rest through the same potential difference.
When a charged particle with charge \(q\) is accelerated through a potential difference \(V\), the work done on the particle by the electric field is equal to the change in its kinetic energy. Starting from rest, the initial kinetic energy is zero. So, the final kinetic energy (\(KE\)) is equal to the work done:
\( KE = \text{Work Done} \)
The work done by the electric field on a charge \(q\) moving through a potential difference \(V\) is given by:
\( \text{Work Done} = qV \)
Thus, the kinetic energy gained by the particle is:
\( KE = qV \)
In this case, both the electron and the proton are accelerated through the same potential difference, \(V = 1000 \, \text{V}\).
- The charge of an electron is \(q_e = -e\), where \(e\) is the elementary charge.
- The charge of a proton is \(q_p = +e\).
The magnitude of the charge is the same for both, \(|q_e| = |q_p| = e\). Since the acceleration is due to the electric field, the work done depends on the magnitude of the charge and the potential difference. Assuming the particles move from lower to higher potential (for proton) or higher to lower potential (for electron) such that they are accelerated, the kinetic energy gained is \(|q|V\).
So, the kinetic energy of the electron is:
\( KE_e = |q_e|V = eV \)
And the kinetic energy of the proton is:
\( KE_p = |q_p|V = eV \)
Therefore, the kinetic energy of both the electron and the proton after being accelerated through the same potential difference is the same.
Comparing the Speed of Electron and Proton
Now let's compare their speeds. The kinetic energy of a particle with mass \(m\) and speed \(v\) is given by:
\( KE = \frac{1}{2}mv^2 \)
From this equation, we can find the speed \(v\):
\( v^2 = \frac{2KE}{m} \)
\( v = \sqrt{\frac{2KE}{m}} \)
Since we know that \(KE_e = KE_p = KE\), the ratio of their speeds will depend on their masses. The mass of an electron (\(m_e\)) is much smaller than the mass of a proton (\(m_p\)). Approximately, \(m_p \approx 1836 \, m_e\).
For the electron, the speed is \(v_e = \sqrt{\frac{2KE}{m_e}}\).
For the proton, the speed is \(v_p = \sqrt{\frac{2KE}{m_p}}\).
Since \(m_e < m_p\), and the kinetic energy \(KE\) is the same for both, the particle with the smaller mass will have a higher speed.
Thus, \(v_e > v_p\).
The speed of the electron will be higher than the speed of the proton.
Analyzing the Given Statements
Let's evaluate each statement based on our findings:
- Statement 1: The kinetic energy of both the particles will be different.
This is incorrect. We found that \(KE_e = KE_p = eV\).
- Statement 2: The speed of the electron will be higher than that of the proton.
This is correct. Since \(KE_e = KE_p\) and \(m_e < m_p\), it follows that \(v_e > v_p\).
- Statement 3: The speed of the proton will be higher than that of the electron.
This is incorrect. The proton has a larger mass and thus a lower speed for the same kinetic energy.
- Statement 4: The speed of the electron and the proton will be equal.
This is incorrect. Their masses are different, leading to different speeds for the same kinetic energy.
Based on the analysis, the statement that the speed of the electron will be higher than that of the proton is correct.
Revision Table: Key Concepts in Particle Acceleration
| Concept | Description | Formula |
|---|---|---|
| Potential Difference (V) | Work done per unit charge to move a charge between two points. | \(V = \frac{W}{q}\) |
| Work Done by Electric Field (W) | Energy transferred to a charge moving through a potential difference. | \(W = qV\) |
| Kinetic Energy (KE) | Energy due to motion. | \(KE = \frac{1}{2}mv^2\) |
| Work-Energy Theorem | Net work done on a particle equals the change in its kinetic energy. | \(W_{net} = \Delta KE\) |
| Electron Charge (qe) | Negative elementary charge. | \(-e\) |
| Proton Charge (qp) | Positive elementary charge. | \(+e\) |
| Electron Mass (me) | Rest mass of an electron. | Approx. \(9.109 \times 10^{-31} \, \text{kg}\) |
| Proton Mass (mp) | Rest mass of a proton. | Approx. \(1.672 \times 10^{-27} \, \text{kg}\) |
Additional Information: Electron vs. Proton Acceleration
When an electron and a proton are accelerated by the same potential difference, it's a fundamental concept in electromagnetism and mechanics. Here are some additional points:
- Charge Magnitude: Both particles have the same magnitude of charge (\(e\)), which is why they gain the same amount of kinetic energy from the same potential difference.
- Mass Difference: The proton is about 1836 times more massive than the electron. This significant mass difference is the primary reason for the difference in their speeds, given the same kinetic energy.
- Direction of Acceleration: If the potential difference is positive (potential increases), a positive charge (proton) is accelerated from lower to higher potential energy (moving opposite to field), but if starting from rest and moving towards lower potential, it gains KE. A negative charge (electron) is accelerated from higher to lower potential energy (moving along the field), gaining KE. The question implies acceleration from rest, so they move in directions that decrease their potential energy, converting potential energy into kinetic energy.
- Relativistic Effects: For acceleration through very high potential differences (thousands or millions of volts), the speeds can approach the speed of light, and relativistic effects become significant. In such cases, the formula \(KE = \frac{1}{2}mv^2\) needs to be replaced with the relativistic kinetic energy formula. However, for 1000 V, the speeds are non-relativistic, and the classical formula is accurate.
Understanding the relationship between potential difference, charge, mass, kinetic energy, and speed is crucial for solving problems involving particle accelerators and motion of charged particles in electric fields.
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