Question

In to a uniform transverse magnetic field, two charged particles having the same mass and charge enter and move in two different circular paths. The ratio of the radii of curvatures of the circular paths is \( 1 : 4 \). What would be the ratio of their respective velocities?

• A. \( \frac{r_1}{r_2} = 4 : 1 \)
• B. \( \frac{r_1}{r_2} = 2 : 1 \)
• C. \( \frac{r_1}{r_2} = 1 : 2 \)
• D. \( \frac{r_1}{r_2} = 1 : 4 \)

Hint:

For a charged particle moving perpendicular to a magnetic field, \( r = \frac{mv}{qB} \), so radius is directly proportional to velocity when \( m, q, B \) are constant.

Answer 1

The Correct Option is D

Step 1: Understand the motion of charged particle.
When a charged particle enters perpendicular to a uniform magnetic field, it moves in a circular path.

Step 2: Use magnetic force as centripetal force.
\[ qvB = \frac{mv^2}{r} \]

Step 3: Rearrange the formula.
\[ r = \frac{mv}{qB} \]

Step 4: Identify constants.
Both particles have same mass \( m \), same charge \( q \), and move in same magnetic field \( B \).

Step 5: Establish proportionality.
\[ r \propto v \]

Step 6: Use given ratio of radii.
\[ r_1 : r_2 = 1 : 4 \]
Therefore:
\[ v_1 : v_2 = 1 : 4 \]

Step 7: Final Answer.
\[ \boxed{1 : 4} \]