Question

Two particles \( X \) and \( Y \) having equal charges are being accelerated through the same potential difference. Thereafter, they enter normally in a region of uniform magnetic field and describe circular paths of radii \( R_1 \) and \( R_2 \) respectively. The mass ratio of \( X \) and \( Y \) is:

• A. \( \left( \frac{R_2}{R_1} \right)^2 \)
• B. \( \left( \frac{R_1}{R_2} \right)^2 \)
• C. \( \frac{R_1}{R_2} \)
• D. \( \frac{R_2}{R_1} \)

Answer 1

The Correct Option is B

To determine the mass ratio of two particles, \( X \) and \( Y \), we consider that they possess identical charges and undergo acceleration through the same potential difference. Subsequently, they enter a uniform magnetic field, traversing circular paths with radii \( R_1 \) for particle \( X \) and \( R_2 \) for particle \( Y \).

The radius \( R \) of a charged particle's circular trajectory in a magnetic field is defined by the formula:

\(R = \frac{mv}{qB}\)

Where:

• \(m\) represents the particle's mass.
• \(v\) is the particle's velocity.
• \(q\) denotes the particle's charge.
• \(B\) indicates the magnetic field strength.

The kinetic energy \(\left( KE \right)\) of a particle accelerated by a potential difference \( V \) is expressed as:

\(\frac{1}{2}mv^2 = qV\)

From this kinetic energy equation, the velocity can be derived as:

\(v = \sqrt{\frac{2qV}{m}}\)

Substituting this velocity expression into the radius formula:

\(R = \frac{\sqrt{\frac{2qV}{m}} \cdot m}{qB} = \frac{m}{qB} \cdot \sqrt{\frac{2qV}{m}}\)

Upon simplification, the radius equation becomes:

\(R = \sqrt{\frac{2Vm}{qB^2}}\)

Given that particles \( X \) and \( Y \) share the same charge \( q \), potential difference \( V \), and magnetic field strength \( B \), with radii \( R_1 \) and \( R_2 \) respectively, we have:

\(R_1 = \sqrt{\frac{2Vm_1}{qB^2}} \quad \text{and} \quad R_2 = \sqrt{\frac{2Vm_2}{qB^2}}\)

Squaring both sides for \( R_1 \) and \( R_2 \):

\(R_1^2 = \frac{2Vm_1}{qB^2} \quad \text{and} \quad R_2^2 = \frac{2Vm_2}{qB^2}\)

To ascertain the mass ratio \(\frac{m_1}{m_2}\), we divide these two squared radius equations:

\(\frac{R_1^2}{R_2^2} = \frac{m_1}{m_2}\)

Therefore, the mass ratio \(\frac{m_1}{m_2}\) is determined by:

\(\frac{m_1}{m_2} = \left( \frac{R_1}{R_2} \right)^2\)

The correct result is:

\(<\left( \frac{R_1}{R_2} \right)^2 \)