Question

A proton and an alpha particle are accelerated through the same potential difference. The ratio of their kinetic energies will be:

• A. \(1:1\)
• B. \(1:2\)
• C. \(2:1\)
• D. \(4:1\)

Hint:

Remember: \[ K=qV \] Greater charge gains greater kinetic energy in same potential difference.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
When a charged particle moves across an electric field, it experiences an electrostatic force that performs mechanical work on it. This work done by the electric field transforms directly into the particle's kinetic energy. If a particle starts from rest, its final kinetic energy depends solely on its net electrical charge and the voltage drop it passes through.
Step 2: Key Formula or Approach:
The work done ($W$) on a particle with charge $q$ accelerated through an electric potential difference $V$ is given by: $$ W = qV $$ According to the work-energy theorem, this work matches the gained kinetic energy ($K$): $$ K = qV $$ Let's define the fundamental charge relationships for both particles: - A proton ($p$) carries a single positive elementary charge: $q_p = e$ - An alpha particle ($\alpha$), which is a helium nucleus ($^4_2\text{He}^{2+}$), carries two protons: $q_\alpha = 2e$
Step 3: Detailed Explanation:
We need to calculate the ratio of the kinetic energy of the proton ($K_p$) to that of the alpha particle ($K_\alpha$). We are given that both particles are accelerated through the exact same potential difference ($V_p = V_\alpha = V$). Let's write out the kinetic energy equations for both particles: - For the proton: $K_p = q_p V = eV$ - For the alpha particle: $K_\alpha = q_\alpha V = 2eV$ Now, find the ratio by dividing the proton's kinetic energy by the alpha particle's kinetic energy: $$ \frac{K_p}{K_\alpha} = \frac{eV}{2eV} $$ Canceling out the common elementary charge factor $e$ and potential difference $V$ simplifies the expression to: $$ \frac{K_p}{K_\alpha} = \frac{1}{2} \implies K_p : K_\alpha = 1 : 2 $$ This evaluation matches option (B).
Step 4: Final Answer:
The ratio of their kinetic energies is 1 : 2.