Problem 15 - Master Resource Book in JEE Main Physics

Question

If in a Thomson's mass spectrograph, the ratio of the electric field and magnetic field, in order to obtain concident parabola of singly ionised and doubly ionised positive ions are \(1: 2\) and \(3: 2\) respectively, then the ratio of masses of particles will be (a) \(3: 1\) (b) \(2: 1\) (c) \(9: 4\) (d) \(9: 2\)

Step-by-Step Solution

Answer

The ratio of masses of particles is \(9:4\) (option c).

Step 1: Understanding the Setup

In a Thomson's mass spectrograph, the paths of ions are bent by electric and magnetic fields. The deflection is such that singly and doubly ionized ions form parabolas when they pass through these fields.

Step 2: Identifying the Given Ratios

We have two given ratios: for singly ionized ions, the electric field to magnetic field ratio is \(1:2\), and for doubly ionized ions, this ratio is \(3:2\). These ratios determine the ions' velocities as they exit the velocity selector.

Step 3: Velocity Relationship with Field Ratios

For a particle with charge \(q\) and mass \(m\), the velocity \(v\) is determined by the equation where the electric force \(qE\) is equal to the magnetic force \(qvB\), so \(v = \frac{E}{B}\). For singly ionized ions, \(v_1 = \frac{1}{2}E/B\) and for doubly ionized ions, \(v_2 = \frac{3}{2}E/B\).

Step 4: Equation for Mass Based on Ion Charge

In the spectrograph, ions with different charges move along different trajectories but coincide at a point depending on their charge \(q\), velocity \(v\), and mass \(m\). The deflection depends inversely on the mass-to-charge ratio \(m/q\).

Step 5: Setting Mass Ratio Equation

For coincidental paths, the effect of the fields must be equivalent, meaning that for singly ionized ions (charge \(e\), velocity \(v_1\)), and doubly ionized ions (charge \(2e\), velocity \(v_2\)), the ratios \((m_1v_1^2)\) and \((m_2v_2^2)\) must match.

Step 6: Calculating Mass Ratio Based on Derived Velocities

Since \(v_1 = \frac{1}{2}E/B\) and \(v_2 = \frac{3}{2}E/B\), setting up the equation \(\frac{m_1(q_1^2)}{v_1^2} = \frac{m_2(q_2^2)}{v_2^2}\) and solving gives \(\frac{m_1}{m_2} = \frac{9}{4}\).

Step 7: Interpreting the Result

The derived mass ratio \(\frac{m_1}{m_2} = \frac{9}{4}\) matches choice (c). Therefore, this is the ratio of the masses of singly and doubly ionized ions.

Key Concepts

Electric and Magnetic Fields

In a mass spectrograph, electric and magnetic fields play a crucial role in analyzing ionized particles. These fields work as a powerful combination to separate ions based on their properties. The electric field exerts a force on charged ions which causes them to accelerate, while the magnetic field bends their path. This bending effect creates a trajectory that can be analyzed to find out various properties of the ions.

For singly and doubly ionized ions, the ratios of electric to magnetic fields are given as essential data points for calculations. The balance and strengths of these fields determine how ions are deflected and their eventual path across the spectrograph, forming distinct parabolic paths based on their mass-to-charge ratios.

Ion Deflection

Ion deflection is the process by which ions are separated based on their properties using external forces. In a mass spectrograph, both electric and magnetic fields work in unison to deflect ions. The degree of deflection helps determine key ion characteristics.

When ions pass through these fields, they experience force, causing them to move in curves or parabolas. The shape and size of this trajectory are determined by several factors:

• The strength of the electric and magnetic fields.
• The mass of the ion.
• The charge of the ion.

In this way, by examining the deflection, scientists can deduce information about the ions, such as identifying isotopes or calculating mass-to-charge ratios.

Mass-to-Charge Ratio

The mass-to-charge ratio (often denoted as \(m/q\)), is a critical factor when analyzing ions in a mass spectrograph. This ratio provides valuable insights into the nature of the ions being studied. The mass-to-charge ratio determines how much an ion will be deflected by the electric and magnetic fields in the spectrograph.

The reason why the mass-to-charge ratio is important is that ions with the same mass but different charges, or the same charge but different masses, will have unique trajectories. This permits the separation and identification of ions with different atomic masses or charge states. Mathematically, if the deflection \(d\) is inversely related to the mass-to-charge ratio, more insights can be drawn by measuring the deflection.

Thomson's Method

Thomson's method, developed by physicist J.J. Thomson, is a pioneering technique in mass spectrometry. This method uses both electric and magnetic fields to measure the properties of ionized particles, making it possible to determine the mass-to-charge ratio of ions accurately.

In this method, ions travel through a velocity selector, which ensures only particular ions with specific velocities enter a further bending path dictated by magnetic fields. As ions of different mass-to-charge ratios follow different paths, one can infer the necessary characteristics of these ions by examining their displacement.

• Firstly, the ions are subjected to a combined effect of both fields, leading to specific paths depending on their charge and mass.
• The degree of bending or parabolic trajectory provides information about the ions' mass-to-charge ratio.

Using Thomson's method, scientists achieved remarkable insights into the composition of ions long before more modern techniques were developed.