Problem 1924 - JEE Physics

Question

In a mass spectrometer used for measuring the masses of ions, the ions are initially accelerated by an ele. potential \(\mathrm{V}\) and then made to describe semicircular paths of radius \(\mathrm{r}\) using a magnetic field \(\mathrm{B}\). If \(\mathrm{V}\) and \(\mathrm{B}\) are kept constant, the ratio [(Charge on the ion) / (mass of the ion)] will be proportional to. (a) \(\left(1 / r^{2}\right)\) (b) \(r^{2}\) (c) \(\mathrm{r}\) (d) \((1 / \mathrm{r})\)

Step-by-Step Solution

Answer

The ratio (charge on the ion) / (mass of the ion) is proportional to \((1/r)\).

Step 1: Understand the working principle of mass spectrometer

: In a mass spectrometer, ions are initially accelerated by an electric potential V, and then they move in a magnetic field B along semicircular paths of radius r. The movement of the ions is governed by the Lorentz force formula, which can be applied to find the relationship between charge, mass, and radius.

Step 2: Apply Lorentz force formula

: The Lorentz force acting on the ion can be given as: \(F_{L} = q(vB)\), where \(F_L\) is the Lorentz force, \(q\) is the charge of the ion, \(v\) is the velocity of the ion, and \(B\) is the magnetic field. Also, the centripetal force acting on the ion can be given as: \(F_{c} = \frac{m v^{2}}{r}\), where \(F_c\) is the centripetal force, \(m\) is the mass of the ion, and \(r\) is the radius of the semicircular path. Since the ion is moving in a semicircular path, the centripetal force and Lorentz force acting on the ion must be equal. Hence, we have: \(q(vB) = \frac{m v^{2}}{r}\)

Step 3: Express velocity in terms of electric potential

: The initial acceleration of the ion under the influence of electric potential can be given as: \(qV = \frac{1}{2}m v^{2}\), where \(V\) is the electric potential. From this equation, the velocity of the ion can be expressed in terms of electric potential, mass, and charge: \(v = \sqrt{2qV/m}\)

Step 4: Substitute the value of velocity in Lorentz force equation

: Now, we will substitute the velocity from the electric potential equation into the Lorentz force and centripetal force equation: \(qB\sqrt{2qV/m} = \frac{m (2qV/m)}{r}\)

Step 5: Simplify the equation and find the ratio

: By simplifying the above equation, we can find the ratio (charge on the ion)/(mass of the ion) is proportional to: \(\frac{q}{m} = \frac{1}{rB}\) Thus, the ratio (charge on the ion)/(mass of the ion) is inversely proportional to the radius. Therefore, the correct answer is: (d) \((1/r)\)

Key Concepts

Lorentz force

The Lorentz force is fundamental in understanding how charged particles behave inside a mass spectrometer. It is the force exerted on a moving charged particle in a magnetic field. This force is described by the equation \(F_{L} = q(vB)\), where \(F_L\) is the Lorentz force, \(q\) is the charge of the particle, \(v\) is its velocity, and \(B\) is the strength of the magnetic field. When ions move through a magnetic field, they experience this force perpendicular to their direction of travel. This causes them to follow a curved path.
Hence, the Lorentz force directly influences the trajectory of ions in a mass spectrometer. When the force is balanced by the centripetal force necessary to maintain the circular motion, you can derive relationships for motion parameters, such as velocity and paths.

Electric potential

Electric potential, often represented as \(V\), is a key player in the initial acceleration of ions in a mass spectrometer. It provides the energy required to propel the ions into motion. The relationship between electric potential and kinetic energy acquired by the ion can be expressed as \(qV = \frac{1}{2}mv^2\), where \(q\) is the charge, \(m\) is the mass, and \(v\) is the velocity.
This equation highlights that the electric potential energy is converted into kinetic energy as the ion speeds up. Understanding this energy conversion is crucial for deriving the velocity of ions, which is then useful for analyzing their motion in a magnetic field.

Centripetal force

Centripetal force is necessary for an ion to maintain its circular path in the presence of a magnetic field. It's given by the formula \(F_{c} = \frac{mv^2}{r}\), where \(F_c\) is the centripetal force, \(m\) is the mass of the ion, \(v\) is its velocity, and \(r\) is the radius of the path.
In a mass spectrometer, the centripetal force is provided by the Lorentz force. For ions moving in semicircular paths, the balance of these forces (\(q(vB) = \frac{mv^2}{r}\)) allows us to solve for unknowns like mass-to-charge ratios. This balance is vital for the analysis of ions based on their unique mass and charge properties.

Magnetic field

The magnetic field, symbolized as \(B\), is integral in steering the ions along curved paths in a mass spectrometer. It works in tandem with the electric potential to achieve this. When charged particles enter the magnetic field, they feel a force perpendicular to their velocity, caused by the Lorentz force.
This field's strength determines the curvature of the ion’s path, directly relating to the radius \(r\) in the semicircular motion equation. Since the ratio of \(\frac{q}{m}\) (charge to mass) is proportional to \(\frac{1}{rB}\), the magnetic field's magnitude helps determine mass and charge characteristics of the ions by altering the path curvature.