Problem 68 - Master Resource Book in JEE Main Physics

Question

In a mass spectrometer used for measuring the masses of ions, the ions are initially accelerated by an electric potential \(V\) and then made to describe semicircular paths of radius \(R\) using a magnetic field B. If \(V\) and \(B\) are kept constant, the ratio \(\left(\frac{\text { charge on the ion }}{\text { mass of the ion }}\right)\) will be proportional to (a) \(1 / R^{2}\) (b) \(R^{2}\) (c) \(R\) (d) \(1 / R\) (e) \(R^{-3}\)

Step-by-Step Solution

Answer

The ratio \( \frac{q}{m} \) is proportional to \( 1/R^2 \). Therefore, the correct answer is (a).

Step 1: Understand the Problem

We need to find the relationship between the ratio \( \frac{q}{m} \), where \( q \) is the charge of an ion and \( m \) is the mass of the ion, with respect to the radius \( R \) of the ion's path in a mass spectrometer. We know the ions are accelerated through an electric potential \( V \) and move through a magnetic field \( B \), both constants of the setup.

Step 2: Relate Kinetic Energy to Electric Potential

The ions are accelerated through a potential difference \( V \), which increases their kinetic energy. The energy acquired by the ion can be expressed as \( qV = \frac{1}{2}mv^2 \), where \( v \) is the velocity of the ion. Therefore, the velocity \( v \) can be derived as \( v = \sqrt{\frac{2qV}{m}} \).

Step 3: Use Magnetic Field to Express the Motion Path

In a magnetic field, the ions follow a semicircular path due to the Lorentz force, which causes a centripetal force. The relationship is \( qvB = \frac{mv^2}{R} \). Rearrange this equation to find the radius \( R \) of the path as \( R = \frac{mv}{qB} \).

Step 4: Substitute for Velocity and Solve

Substitute \( v = \sqrt{\frac{2qV}{m}} \) into the equation for radius: \( R = \frac{m(\sqrt{\frac{2qV}{m}})}{qB} = \frac{\sqrt{2m^2V}}{qB} \). Simplifying gives \( R = \frac{\sqrt{2mV}}{qB} \).

Step 5: Express Charge-to-Mass Ratio in terms of Radius

To find the charge-to-mass ratio, start by squaring both sides of the expression for \( R \): \( R^2 = \frac{2mV}{(qB)^2} \). Rearrange the expression to isolate \( \frac{q}{m} \): \( q/m = \frac{2V}{B^2 R^2} \).

Step 6: Determine Correct Option

The expression \( q/m = \frac{2V}{B^2 R^2} \) shows that \( \frac{q}{m} \) is proportional to \( 1/R^2 \) since \( V \) and \( B \) are constants. Therefore, the correct answer is option (a) \( 1/R^2 \).

Key Concepts

Ionic Charge

In the context of a mass spectrometer, an ion's charge plays a crucial role in determining its behavior when subjected to electric and magnetic fields. The ionic charge, denoted as \( q \), is a measure of the excess or deficit of electrons in an atom or molecule. This charge affects how ions respond to external forces like those observed in the electric and magnetic fields of a mass spectrometer.

When an ion enters a mass spectrometer, its charge affects its kinetic energy and its trajectory within the device. The charge directly influences the force the ion experiences in a magnetic field, according to the Lorentz force law. This law is given by the equation \( F = qvB \), where \( F \) is the force, \( v \) is velocity, and \( B \) is the magnetic field.

Here are key aspects concerning ionic charge in a mass spectrometer:

• The greater the charge, the more it influences the ion’s motion in a field.
• Ions with higher charges usually have higher kinetic energy when accelerated through an electric potential.

Understanding the ionic charge is crucial for determining how ions will travel within the spectrometer and how their paths will differ based on mass and charge.

Magnetic Field

A magnetic field in a mass spectrometer is essential for analyzing and separating ions based on their mass-to-charge ratios. The magnetic field, represented as \( B \), interacts with the charged ions, causing them to follow curved paths. This bending of the ion paths depends on both their charge \( q \) and velocity \( v \).

The relationship between the magnetic field and ion motion is given by the centripetal force equation that results in circular motion: \( qvB = \frac{mv^2}{R} \). Here \( R \) is the radius of the ion's path. This equation tells us how the strength of the magnetic field affects the curvature of the ion's path.

Key points about magnetic fields in mass spectrometers include:

• A stronger magnetic field results in tighter ion paths, affecting the path radius \( R \).
• The magnetic field helps segregate ions based on their mass-to-charge ratio by inducing different path radii.

The manipulation of magnetic fields allows scientists to accurately determine the properties of various ions, making it a critical component in the use of mass spectrometers.

Electric Potential

Electric potential in a mass spectrometer is utilized to accelerate ions before they enter the magnetic field. Represented by \( V \), the electric potential is essentially the electrical energy per unit charge provided to an ion. This increase in energy results in a corresponding increase in the ion's velocity, \( v \), which is crucial for its subsequent movement through the magnetic field.

When ions are accelerated through an electric potential difference, their gain in kinetic energy can be expressed as: \( qV = \frac{1}{2}mv^2 \). Here, \( q \) is the ion's charge, \( m \) is its mass, and \( V \) is the potential difference.

Important aspects of electric potential in the context of a mass spectrometer include:

• Higher electric potential increases the kinetic energy of the ion.
• The kinetic energy gained at this stage directly affects the radius of the ion’s path in the magnetic field.
• Consistent electric potential ensures accurate and repeatable ion analysis.

In a mass spectrometer, regulating the electric potential enables precise control over ion speeds, thereby affecting how they are separated according to their mass-to-charge ratio.