Problem 1911 - JEE Physics

Question

1: A charged particle moves in a uniform mag. field. The velocity of the particle at some instant makes an acute angle with the mag. field. The path of the particle will be (a) A straight line (b) A circle (c) A helix with uniform pitch (d) A helix with non-uniform pitch

Step-by-Step Solution

Answer

The path of the charged particle will be a \(helix\) with a uniform pitch when its velocity makes an acute angle with the magnetic field. This is due to the combination of motion along the magnetic field (straight line) and the motion perpendicular to the magnetic field (circle) caused by the Lorentz force acting on the particle.

Step 1: Identify the force acting on the charged particle

: When a charged particle with charge q moves with a velocity \(\vec{v}\) in a magnetic field \(\vec{B}\), it experiences a force called the Lorentz force. This force can be calculated using the following formula: \( \vec{F} = q( \vec{v} \times \vec{B} ) \) where \(\vec{F}\) is the Lorentz force and \(\times\) denotes the cross product.

Step 2: Analyze the motion of the particle in the magnetic field

: The force acting on the particle is always perpendicular to its velocity (\(\vec{v}\)). Therefore, when the charged particle enters the magnetic field at an acute angle, the trajectory can be divided into two components. The first component is along the magnetic field and the second component is perpendicular to the magnetic field.

Step 3: Determine the path along the magnetic field

: Since the force acting on the particle is always perpendicular to its velocity, there will be no force acting on a particle moving in the direction of the magnetic field. Thus, along the magnetic field, the particle will continue moving with a constant velocity, which means that it would form a straight line along the magnetic field direction.

Step 4: Determine the path perpendicular to the magnetic field

: Perpendicular to the magnetic field, the charged particle experiences a centripetal force due to the Lorentz force, which causes the particle to move in a circular path. The radius of this circle can be determined by setting the centripetal force equal to the magnetic force: \( \frac{mv^2}{r} = qvB \) where m is the mass of the particle, v is its velocity perpendicular to the magnetic field, and r is the radius of the circular path.

Step 5: Combine both paths to find the complete path

: Since the motion of the particle is a combination of the motion along the magnetic field (straight line) and the motion perpendicular to the magnetic field (circle), the complete path of the charged particle will be a helix with a uniform pitch. Thus, the correct option is: (c) A helix with uniform pitch

Key Concepts

Lorentz Force

The Lorentz force is a fundamental concept in electromagnetism. It describes the force that a charged particle experiences when it moves within a magnetic field. Imagine a charged particle, like an electron or a proton, drifting along with a certain velocity. When this particle enters a magnetic field, it encounters a force, explained by the formula:

• \( \vec{F} = q( \vec{v} \times \vec{B} ) \)

This equation tells us that the force \( \vec{F} \) depends on the charge \( q \), the velocity \( \vec{v} \) of the particle, and the magnetic field \( \vec{B} \). The cross product \((\times)\) signifies that this force is perpendicular to both the velocity of the particle and the magnetic field. This perpendicular relationship is the reason why charged particles have such unique and interesting paths in magnetic fields.

Charged Particle Motion

The motion of a charged particle within a magnetic field is unique because of the Lorentz force acting perpendicular to its velocity. When a charged particle, like an electron, enters a magnetic field with some initial velocity that is not aligned with the field, its path becomes quite distinct. There are two key components:

• One parallel to the magnetic field: In this direction, the particle moves straight, as no force acts upon it in this direction. Thus, it maintains a constant velocity.
• One perpendicular to the magnetic field: In this direction, the particle encounters the Lorentz force that acts as a centripetal force, making the particle move in a circular path.

By combining both of these motions, the particle follows a helical path. This is a spiral-like shape formed due to the coexistence of linear and circular motion.

Helical Path

The helical path of a charged particle is a result of the combined motions—linear along the magnetic field and circular due to the Lorentz force. To visualize a helix, think of the shape of a spring or a spiral staircase. In physics, a helix is characterized by uniform pitch. The pitch refers to the distance between successive turns along the axis of the helix.

• Uniform Pitch: The uniform pitch means all turns of the spiral are equally spaced, indicating a constant velocity component parallel to the field.

The charged particle's velocity component along the magnetic field ensures that while its path coils around, it consistently progresses in one direction, creating the helical trajectory.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic force is the same at all points. Imagine a magnetic field that looks like evenly spaced parallel lines. In such a field, a charged particle like an electron will experience a consistent force as it moves within it.

• Due to this consistency, the path of the particle, influenced by the Lorentz force, becomes predictable and regular.
• The regularity ensures that any circular path component will have a constant radius, keeping the pitch of the helical path uniform.

The importance of uniformity is that it provides a stable environment allowing scientists to predict particle behaviour easily. This predictability is crucial in applications like particle accelerators and various electromagnetic devices.