Question

Two particles A and B have equal charges but different masses $M_A$ and $M_B$. After being accelerated through the same potential difference, they enter a region of uniform magnetic field and describe paths of radii $R_A$ and $R_B$ respectively. Then $M_A : M_B$ is

• A. $\dfrac{R_A}{R_B}$
• B. $\dfrac{R_B}{R_A}$
• C. $\left(\dfrac{R_A}{R_B}\right)^2$
• D. $\left(\dfrac{R_B}{R_A}\right)^2$

Hint:

For particles accelerated by same voltage, radius in magnetic field varies as $\sqrt{m}$.

Answer 1

The Correct Option is C

Step 1: Velocity after acceleration through same potential difference.
When a charged particle is accelerated through potential difference $V$:
\[ \frac{1}{2}mv^2 = qV \Rightarrow v = \sqrt{\frac{2qV}{m}} \]

Step 2: Radius of circular path in magnetic field.
\[ R = \frac{mv}{qB} \] Substitute $v$:
\[ R = \frac{m}{qB}\sqrt{\frac{2qV}{m}} = \frac{1}{B}\sqrt{\frac{2mV}{q}} \]

Step 3: Relation between radius and mass.
\[ R \propto \sqrt{m} \Rightarrow m \propto R^2 \]

Step 4: Ratio of masses.
\[ \frac{M_A}{M_B} = \left(\frac{R_A}{R_B}\right)^2 \]

Step 5: Conclusion.
The correct ratio is $\left(\dfrac{R_A}{R_B}\right)^2$.