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 Question:
We are studying the motion of a charged particle in a uniform electric field.
The particle starts from rest and travels a fixed distance $d$ under the influence of the electric field.
We need to find how the travel time depends on the mass of the particle.

Step 2: Key Formula or Approach:
The electrostatic force acting on a charge $q$ in a field $\mathbf{E}$ is $F = qE$.
Using Newton's second law, the acceleration of the particle is $a = \frac{qE}{m}$.
Using the equation of motion for a particle starting from rest ($u = 0$):
\[ d = \frac{1}{2} a t^2 \]

Step 3: Detailed Explanation:

• Let us express the distance $d$ in terms of mass and time:
\[ d = \frac{1}{2} \left( \frac{qE}{m} \right) t^2 \]
• Solving for the travel time $t$:
\[ t^2 = \frac{2md}{qE} \implies t = \sqrt{\frac{2md}{qE}} \]
• Since the parameters $q$, $E$, and $d$ are constant for both cases:
\[ t \propto \sqrt{m} \]
• Therefore, the ratio of the travel times for the two different masses $m_1$ and $m_2$ is:
\[ \frac{t_1}{t_2} = \sqrt{\frac{m_1}{m_2}} \]

Step 4: Final Answer:
The ratio $\frac{t_1}{t_2}$ is equal to $\sqrt{\frac{m_1}{m_2}}$.