Question

Two particles $A$ and $B$ having same mass have charge $+q$ and $+4q$ respectively. When they are allowed to fall from rest through same electric potential difference, the ratio of their speeds $V_A$ to $V_B$ will become

• A. $1 : 2$
• B. $2 : 1$
• C. $1 : 4$
• D. $4 : 1$

Hint:

Since kinetic energy scales linearly with charge ($K.E. \propto q$) for a constant voltage, particle $B$ will gain exactly 4 times the kinetic energy of particle $A$. Because velocity scales with the square root of kinetic energy ($v \propto \sqrt{K.E.}$), particle $B$ will move at $\sqrt{4} = 2$ times the speed of particle $A$, giving the ratio $1 : 2$ instantly!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given two particles, $A$ and $B$, which have identical masses ($m_A = m_B = m$) but different electrical charges ($q_A = +q$ and $q_B = +4q$). Both particles start from rest ($u = 0$) and accelerate through the exact same electrical potential difference $V$. We need to find the ratio of their final speeds, $V_A : V_B$.

Step 2: Key Formula or Approach:
According to the work-energy theorem, the work done by the electrostatic force on a charged particle moving through a potential difference $V$ is converted entirely into its kinetic energy: $$W = qV = \Delta K.E. = \frac{1}{2}mv^2 - 0$$ Isolating the velocity $v$ from this equality shows that: $$v = \sqrt{\frac{2qV}{m}}$$ Since the mass $m$ and potential difference $V$ are identical for both particles, the final speed is directly proportional to the square root of the charge: $v \propto \sqrt{q}$.

Step 3: Detailed Explanation:
Let's set up the ratio of the final velocities for particles $A$ and $B$ using our proportional relationship: $$\frac{V_A}{V_B} = \sqrt{\frac{q_A}{q_B}}$$ Substitute the given charge values ($q_A = q$ and $q_B = 4q$) into the equation: $$\frac{V_A}{V_B} = \sqrt{\frac{q}{4q}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$$ This yields a final speed ratio of $1 : 2$.

Step 4: Final Answer:
The ratio of their speeds is $1 : 2$, which corresponds to option (A).