Problem 32 - Master Resource Book in JEE Main Physics

Question

A proton of mass \(1.67 \times 10^{-27} \mathrm{~kg}\) and charge \(1.6 \times 10^{-19} \mathrm{C}\) is projected with a speed of \(2 \times 10^{6} \mathrm{~ms}^{-1}\) at an angle of \(60^{\circ}\) to the \(x\)-axis. If a uniform magnetic field of \(0.104 \mathrm{~T}\) is applied along \(y\)-axis, the path of proton is (a) a circle of radius \(=0.2 \mathrm{~m}\) and time period \(=2 \pi \times 10^{-7} \mathrm{~s}\) (b) a circle of radius \(=0.1\) mand time period \(=2 \pi \times 10^{-7}\) s (c) a helix of radius \(0.1 \mathrm{~m}\) and time period \(=2 \pi \times 10^{-7} \mathrm{~s}\) (d) a helix of radius \(0.2 \mathrm{~m}\) and time period \(=2 \pi \times 10^{-7} \mathrm{~s}\)

Step-by-Step Solution

Answer

Option (c): a helix of radius \(0.1 \mathrm{~m}\) and time period \(2 \pi \times 10^{-7} \mathrm{~s}\).

Step 1: Analyze Velocity Components

The proton's velocity has two components due to the angle of projection. Calculate these components: \( v_{x} = v \cos \theta = (2 \times 10^6 \text{ m/s}) \cos 60^{\circ} = 1 \times 10^6 \text{ m/s}\) and \( v_{y} = v \sin \theta = (2 \times 10^6 \text{ m/s}) \sin 60^{\circ} = \sqrt{3} \times 10^6 \text{ m/s}\).

Step 2: Determine the Type of Path

Since the magnetic field is along the y-axis, any velocity component along the y-axis remains unaffected by the magnetic field. Only the x-component of velocity will result in circular motion. As there is non-zero movement along both the x and y directions, the path is helical.

Step 3: Calculate Helical Path Radius

The radius of the circular component of the helical path is given by \( R = \frac{m v_{x}}{q B} \). Substituting the known values: \( R = \frac{(1.67 \times 10^{-27} \text{ kg}) (1 \times 10^6 \text{ m/s})}{(1.6 \times 10^{-19} \text{ C})(0.104 \text{ T})} \approx 0.1 \text{ m} \).

Step 4: Calculate the Time Period of the Circulation

The time period \( T \) for one complete revolution in the magnetic field can be calculated using \( T = \frac{2 \pi m}{q B} \). Substituting the known values: \( T = \frac{2 \pi (1.67 \times 10^{-27} \text{ kg})}{(1.6 \times 10^{-19} \text{ C})(0.104 \text{ T})} \approx 2\pi \times 10^{-7} \text{ s} \).

Key Concepts

Helical Motion

When a charged particle like a proton enters a magnetic field at an angle, its path becomes a helical one. This is because the velocity of the proton can be divided into two components: one parallel to the magnetic field, and one perpendicular. The perpendicular component causes the proton to undergo circular motion due to the magnetic force, forming the circular part of the helix. The parallel component, however, continues in a straight line without any influence from the magnetic field, causing the forward "step" or "pitch" of the helix.
Here are the key points to understanding helical motion in a magnetic field:

• A helical path occurs when a particle has velocity components both parallel and perpendicular to a magnetic field.
• The circular motion results from the perpendicular component of velocity, while the forward motion results from the parallel component.
• The size of the circular path, or the radius, is determined by the strength of the perpendicular velocity component and the magnetic field itself.
• The "pitch" or step length of the helix is determined by the parallel velocity component.

Understanding these concepts helps in visualizing how particles behave in magnetic fields, providing insight into many real-world applications like cyclotrons and magnetically confined plasma in fusion reactors.

Proton Trajectory

The path of a proton in a magnetic field depends greatly on its initial velocity and the angle of entry with respect to magnetic field lines. In our example, the proton is projected with a velocity at an angle of \(60^{\circ}\) to the \(x\)-axis in a uniform magnetic field aligned along the \(y\)-axis.
Here’s a breakdown of how the trajectory is determined:

• Velocity Components: The velocity divides into two components because of the angle of release: \(v_{x} = v \cos \theta\) and \(v_{y} = v \sin \theta\). The component along \(x\) influences the radius of the circular trajectory, while the \(y\) component aids in the linear progression of the helix.
• Magnetic Force Influence: The proton experiences the Lorentz force, altering the trajectory depending on the velocity's direction and magnitude relative to the field lines.
• Path Formation: With non-zero components along \(x\) and \(y\), the proton moves in a helical path, characterized by circular motion along one plane and straight-line motion in another direction.

In essence, proton trajectory in a magnetic field is a fascinating display of predicting motion using physics principles, showcasing the interplay of charged particles and magnetic forces.

Lorentz Force

The Lorentz force is the central reason why charged particles exhibit helical motion in a magnetic field. This force influences a charged particle when it moves through a magnetic field, determining its path, speed, and the resultant trajectory.
Key points about the Lorentz force include:

• Definition: The force exerted on a charged particle by magnetic and electric fields, given by \(F = q(v \times B)\), where \(q\) is the charge, \(v\) is the velocity, and \(B\) is the magnetic field.
• Directionality: The direction of the force is perpendicular to both the velocity and magnetic field, described by the right-hand rule.
• Effect on Motion: The perpendicular nature of the Lorentz force causes the particle to change direction constantly, resulting in circular or helical paths rather than linear ones.
• Magnitude: The force depends on the magnitude of the charge and both the speed of the particle and the strength of the magnetic field.

The concept of the Lorentz force is fundamental in understanding how particles move in magnetic fields and is pivotal in applications such as the design of mass spectrometers and cyclotrons.

Magnetic Field Interactions

Interactions with magnetic fields are crucial in determining the motion of charged particles. When a particle like a proton enters a magnetic field, various interactions dictate its trajectory and behavior.
Highlights of magnetic field interactions are:

• Nature of the Field: A uniform magnetic field exerts consistent forces on the charged particle, creating predictable paths like circles or helices.
• Velocity Decomposition: A charged particle's velocity splits into components parallel and perpendicular to the field, influencing the path's nature.
• Application of the Right-Hand Rule: This rule helps determine the direction of motion of a charged particle in a magnetic field, essential for understanding trajectory.
• Practical Applications: These interactions are used in various technologies, including particle accelerators and electromechanical devices, showcasing their practical significance.

Understanding magnetic field interactions is vital for grasping the dynamics of particles in fields and serves as the basis for numerous technologies and scientific investigations.