An electron and a proton, both starting from rest, are accelerated through the same potential difference of 100 kV. Their final speeds are denoted by V_e for the electron and V_p for the proton. Which of the following relations between V_e and V_p is correct under classical (non relativistic) consideration?

Introduction / Context:

Charged particles like electrons and protons gain kinetic energy when they are accelerated through an electric potential difference. This question compares the speeds of an electron and a proton that both start from rest and are accelerated through the same potential difference of 100 kV. The key idea is to relate the gain in kinetic energy to the particle mass and then compare the resulting speeds.

Given Data / Assumptions:

• Electron charge magnitude and proton charge magnitude are equal to e.
• Both particles start from rest.
• Both experience the same potential difference V = 100 kV.
• Relativistic effects are neglected for the purpose of this conceptual question.

Concept / Approach:

When a charge q moves through a potential difference V, the work done by the electric field on the charge is q * V. This work appears as kinetic energy if the particle starts from rest. Therefore, the kinetic energy gained by each particle is KE = q * V. Since the electron and proton have the same charge magnitude e, they gain the same kinetic energy. The speed v of a particle with mass m and kinetic energy KE is given by KE = (1 / 2) * m * v^2. For the same KE, a lighter particle must have a higher speed than a heavier particle.

Step-by-Step Solution:

Verification / Alternative check:

We can compare the speeds symbolically by equating e * V = (1 / 2) * m * v^2 and solving for v to get v = sqrt(2 * e * V / m). For the electron and proton, the numerator 2 * e * V is the same, but the denominator mass m differs by a factor of about 1836 (the mass ratio of proton to electron). Thus V_e / V_p = sqrt(m_p / m_e) which is much greater than 1, confirming that the electron is much faster.

Why Other Options Are Wrong:

Option B: V_e < V_p contradicts the dependence v proportional to 1 / sqrt(m). A heavier particle cannot be faster if both receive the same kinetic energy.

Option C: V_e = V_p would require the masses to be equal, which is not true for electrons and protons.

Option D: The speeds can be compared using basic formulas, so it is not true that they cannot be determined.

Common Pitfalls:

Students sometimes focus on the fact that q and V are the same and incorrectly assume that the speeds must also be the same. The crucial missing step in that reasoning is the role of mass in the kinetic energy expression. Remember that equal kinetic energies do not imply equal speeds when the masses are different. Lighter particles move faster for the same kinetic energy.

Final Answer:

The correct relation is V_e > V_p.