Problem 3 - Master Resource Book in JEE Main Physics

Question

The working principle of the mass spectrograph is that for a given combination of accelerating potential and magnetic field, the ion beam (with charge \(q\) and mass \(M\) ) to be collected at different positions of ion collectors will depend upon the value of (a) \(\sqrt{\frac{q}{M}}\) (b) \(\left(\frac{q}{M}\right)^{2}\) (c) \(\frac{q}{M}\) (d) \(q M\)

Step-by-Step Solution

Answer

The correct answer is (a) \( \sqrt{\frac{q}{M}} \).

Step 1: Understand the Principle

The mass spectrograph works on the principle of deflecting ion beams in a magnetic field. The ions are accelerated by a potential and then pass through a magnetic field, where they are deflected. The deflection depends on the mass to charge ratio \( \frac{M}{q} \).

Step 2: Relate Ion Deflection to Charge and Mass

In a magnetic field, the force on the ion is given by the Lorentz force \( F = qvB \), where \( v \) is the velocity and \( B \) is the magnetic field. The ion moves in a circular path with radius \( r \), satisfying \( qvB = \frac{Mv^2}{r} \). Rearranging gives \( \frac{r}{v} = \frac{M}{qB} \).

Step 3: Deriving Velocity in Relation to Potential

The velocity \( v \) is obtained from the kinetic energy given to the ions by the accelerating potential \( V \): \( \frac{1}{2}Mv^2 = qV \). Solving for \( v \) gives \( v = \sqrt{\frac{2qV}{M}} \).

Step 4: Substitute to Find Deflection Dependency

Substitute \( v \) from Step 3 into the equation \( \frac{r}{v} = \frac{M}{qB} \). We have \( r = \frac{Mv}{qB} = \frac{M}{qB} \sqrt{\frac{2qV}{M}} \). Simplify this to \( r = \sqrt{\frac{2MV}{qB^2}} \) which shows that \( r \propto \sqrt{\frac{M}{q}} \). Therefore, the dependency on the value is \( \sqrt{\frac{q}{M}} \).

Step 5: Identify the Correct Option

Considering the derived proportional relationship, the ion beam's position on the ion collector is directly related to the expression \( \sqrt{\frac{q}{M}} \). Thus, the correct answer is option (a) \( \sqrt{\frac{q}{M}} \).

Key Concepts

Mass-to-Charge Ratio

The mass-to-charge ratio is a fundamental concept in mass spectrometry, determining how ions behave in a magnetic field. It is represented by the expression \( \frac{M}{q} \), where \( M \) is the mass of the ion and \( q \) is its charge.
The significance of this ratio is in how it affects the path of ions in the mass spectrograph.- When ions with different mass-to-charge ratios are subjected to the same magnetic and electric fields, their paths differ.- This is because heavier ions or those with lower charge experience less deflection.Thus, the mass-to-charge ratio directly influences where ions will land on the ion detector, allowing us to separate and analyze ions based on their masses.

Lorentz Force

The Lorentz force is critical in understanding how magnetic fields influence moving charged particles, such as ions. It is described by the formula \( F = qvB \), where:- \( F \) is the force acting on the charged ion.- \( q \) is the charge.- \( v \) is the velocity.- \( B \) is the magnetic field strength.

In the mass spectrograph, this force causes ions to travel in curved paths. Ions moving through the magnetic field perpendicular to its lines are deflected.
This deflection depends on the mass-to-charge ratio, as well as the strength of the magnetic field. The relationship between these factors determines how much an ion's path curves, playing a key role in separating different ions for analysis.

Ion Deflection

Ion deflection in a magnetic field is a core principle for separating ions based on their mass-to-charge ratios. When ions enter a magnetic field, they experience the Lorentz force, causing them to deflect.
The radius of their curved path \( r \) is given by \( r = \frac{Mv}{qB} \).

Here, deflection helps in distinguishing ions in the spectrograph:- Ions with a larger mass or lower charge deflect less, following a larger circular path.- Conversely, ions with a smaller mass or higher charge deflect more, following a tighter circular path.

By quantifying their deflection, we can effectively separate and identify ions in a sample, crucial for analytical applications in chemistry and physics.

Accelerating Potential

The accelerating potential is essential for giving ions sufficient speed in a mass spectrograph. It denotes the electric potential \( V \) used to accelerate ions before they enter the magnetic field. The kinetic energy imparted to the ions is defined by \( \frac{1}{2}Mv^2 = qV \). Solving for velocity, we get \( v = \sqrt{\frac{2qV}{M}} \).

This velocity determines:- The energy and speed at which ions move into the magnetic field.- The extent of their deflection, as velocity directly affects the Lorentz force applied to them.The interplay between accelerating potential, ion velocity, and magnetic field strength dictates the path ions take, and therefore their position on the detector. By carefully choosing the accelerating potential, we can optimize the separation of ions in the mass spectrograph.