Problem 22 - Master Resource Book in JEE Main Physics

Question

In a mass spectrograph, an ion \(X\) of mass number 24 and charge \(+e\) and another ion \(Y\) of mass number 22 and charge+ \(2 e\) enter in a perpendicular magnetic field with the same velocity. The ratio of the radii of the circular path in the field will be (a) \(\frac{11}{22}\) (b) \(\frac{11}{2}\) (c) \(\frac{22}{11}\) (d) \(\frac{24}{11}\)

Step-by-Step Solution

Answer

The radius ratio is \( \frac{24}{11} \), so the correct option is (d).

Step 1: Understand the Problem

We need to find the ratio of the radii of the circular paths of two ions, X and Y, in a magnetic field. Each ion has a mass number and charge, and we need to consider these to determine the paths.

Step 2: Use the Formula for Radius in Magnetic Field

The radius \( r \) of a charged particle in a magnetic field is given by the formula \( r = \frac{mv}{qB} \), where \( m \) is the mass, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field. Since both ions enter with the same velocity \( v \), and in the same magnetic field \( B \), these variables can be treated as constants when comparing the two ions.

Step 3: Calculate the Effective Masses and Charges

For ion \( X \), the mass is proportional to its mass number 24, and its charge is \( +e \). For ion \( Y \), the mass is proportional to its mass number 22, and the charge is \( +2e \). This helps determine the radii of their paths using their mass-to-charge ratios.

Step 4: Determine the Ratio of Radii

Using the formula from Step 2, the ratios of the radii \( r_X \) and \( r_Y \) can be calculated as follows:\[ \frac{r_X}{r_Y} = \frac{\frac{m_X v}{q_X B}}{\frac{m_Y v}{q_Y B}} = \frac{m_X q_Y}{m_Y q_X} \]Substituting the values, we get:\[ \frac{r_X}{r_Y} = \frac{24 \times 2e}{22 \times e} = \frac{48}{22} = \frac{24}{11} \]

Step 5: Conclude with the Given Options

Among the given options, the ratio we calculated \( \frac{24}{11} \) matches option (d). Therefore, the correct answer is option (d).

Key Concepts

Mass-to-Charge Ratio

The mass-to-charge ratio is a fundamental concept in mass spectrometry. It describes the relationship between the mass of an ion and its charge. In mass spectrographs, this ratio helps determine how ions travel when exposed to a magnetic field.

• For ion X: Mass number = 24, Charge = +e
• For ion Y: Mass number = 22, Charge = +2e

Understanding this ratio is crucial because ions with different mass-to-charge ratios will have different trajectories. Ions with a larger mass-to-charge ratio will generally move in larger circles in a uniform magnetic field, affecting how they are separated in a mass spectrograph. The mass-to-charge ratio can be expressed as:\[ \frac{m}{q} \]where \( m \) is mass and \( q \) is charge.

Magnetic Field

A magnetic field plays a critical role in a mass spectrograph. It is used to bend the path of charged particles such as ions. The strength and direction of the magnetic field influence the trajectory and separation of ions.

• The magnetic field is denoted as \( B \) in calculations.
• It causes charged particles to move in curved paths, whose radius depends on the particle's velocity, charge, and mass.

The interaction of ions with the magnetic field leads to circular motion, enabling the separation and analysis of different ions based on their mass-to-charge ratio.

Circular Motion

When charged particles like ions enter a perpendicular magnetic field, they undergo circular motion. This is due to the Lorentz force, which acts perpendicular to the velocity of the ions, altering their straight-line path into a circle.

• The radius of the circle, \( r \), is determined by the particle's mass, charge, velocity, and the magnetic field.
• The formula for the radius is: \( r = \frac{mv}{qB} \)

In our specific example, both ions have the same initial velocity. Their paths differ based on their mass and charge, leading to different radii for ions X and Y as they move through the magnetic field.

Ion Separation

Ion separation in a mass spectrograph relies on the distinct trajectories ions follow when subjected to a magnetic field. Different ions diverge based on their mass-to-charge ratios, which helps in distinguishing between various isotopes or molecular fragments.

• Ions with higher mass-to-charge ratios follow larger orbital paths.
• The extent of separation aids in identifying them based on the radius of their circular paths in the magnetic field.

In the example problem, ion X and ion Y are separated due to their different mass-to-charge characteristics, providing a mechanism to calculate their relative path radii.

Charge of Ions

The charge of an ion significantly influences how it reacts in a magnetic field. This charge, typically denoted as \(+e\) or \(+2e\), directly impacts the force on the particle and, hence, the radius of its circular motion.

• Ion X: Charge is \(+e\)
• Ion Y: Charge is \(+2e\)

The greater the charge, the stronger the interaction with the magnetic field, and consequently, the smaller the circular path. Therefore, the charge magnitude is a pivotal factor in determining the path that an ion will follow, dictating its final position in a mass spectrograph analysis.