Question

A particle of mass $m_1$ and electric charge $q$ starts from rest under the influence of a uniform external electric field $\mathbf{E}$ to travel a distance $d$ in time $t_1$. If the particle had mass $m_2$, it would take time $t_2$ to travel the same distance. What is the ratio $\frac{t_1}{t_2}$?

• A. $\sqrt{\frac{m_1}{m_2}}$
• B. $\sqrt{\frac{m_2}{m_1}}$
• C. $\frac{m_2}{m_1}$
• D. $\frac{m_1}{m_2}$

Hint:

For constant force motion starting from rest, acceleration is inversely proportional to mass ($a \propto 1/m$).
Since $d = \frac{1}{2}at^2$, the time taken to travel a fixed distance is proportional to the square root of mass ($t \propto \sqrt{m}$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The particle undergoes constant acceleration motion starting from rest.
Key Formula or Approach:
Force \( F = qE = ma \implies a = \frac{qE}{m} \).
Distance \( d = \frac{1}{2}at^{2} \).

Step 2: Detailed Explanation:
For a fixed distance \( d \):
\[ d = \frac{1}{2} \left( \frac{qE}{m} \right) t^{2} \implies t^{2} = \frac{2md}{qE} \]
Since \( q, E, d \) are constants for both cases:
\[ t^{2} \propto m \implies t \propto \sqrt{m} \]
Taking the ratio for masses \( m_{1} \) and \( m_{2} \):
\[ \frac{t_{1}}{t_{2}} = \sqrt{\frac{m_{1}}{m_{2}}} \]

Step 3: Final Answer:
The ratio is \( \sqrt{\frac{m_{1}}{m_{2}}} \).