Problem 6 - Master Resource Book in JEE Main Physics

Question

A uniform electric and magnetic fields are produced pointing in the same direction. If an electron is projected with its velocity pointing in the same direction. \(\quad\) [NCERT Exemplar] (a) The electron velocity will decrease in magnitude (b) The electron velocity will increased in magnitude (c) neither (a) nor (b) (d) None of the above

Step-by-Step Solution

Answer

Neither (a) nor (b), as the answer depends on the alignment of the electric field with the electron's velocity.

Step 1: Understand the Context

In this problem, an electron moves in the presence of both an electric field (E) and a magnetic field (B), with both fields aligned in the same direction as the velocity of the electron. Our goal is to evaluate how these fields affect the electron's velocity.

Step 2: Analyze the Electric Field's Effect

The electric field ("]) exerts a force on the electron given by \( \vec{F}_{e} = q \vec{E} \), where \( q \) is the charge of the electron. This force is parallel to the field direction, affecting the electron's speed by accelerating or decelerating it, depending on the relative directions of the field and motion.

Step 3: Analyze the Magnetic Field's Effect

The magnetic force on a charged particle is given by \( \vec{F}_{b} = q (\vec{v} \times \vec{B}) \). Since the velocity \( \vec{v} \) and the magnetic field \( \vec{B} \) are parallel, the cross product is zero: \( \vec{F}_b = q(\vec{v} \times \vec{B}) = 0 \). Thus, the magnetic field does not affect the electron's velocity magnitude.

Step 4: Combine Effects and Conclude

Since the magnetic field exerts no force, only the electric field influences the electron's velocity. If the electric field direction opposes the electron's motion, its velocity will decrease; otherwise, it will increase. As only the electric field has potential to change the velocity, it does so dependent on its alignment with the electron's velocity.

Key Concepts

Electric Field

Electric fields are regions surrounding a charge that exert force on other charges. They are vector fields and have both a magnitude and a direction. When an electric field acts on a charged particle, such as an electron, it applies a force given by the equation \[ \vec{F}_e = q \vec{E} \] where \( q \) is the charge of the particle and \( \vec{E} \) is the electric field. For electrons, which have a negative charge, the direction of the force is opposite to the direction of the field.
This force can change the speed of the electron depending on the alignment of the field and the electron's motion. If the field is aligned with the motion, the electron speeds up. If opposite, it slows down.
Thus, electric fields are crucial in determining how charged particles move, influencing their velocity based on field direction.

Magnetic Field

A magnetic field is created by moving charges or magnetic materials and influences other moving charges. Unlike electric fields that affect both stationary and moving charges, magnetic fields exert forces only on moving charges.
The force exerted by a magnetic field on a charged particle moving with velocity \( \vec{v} \) is given by the Lorentz force part: \[ \vec{F}_b = q (\vec{v} \times \vec{B}) \] where \( q \) is the charge and \( \vec{B} \) is the magnetic field.
In cases where the velocity and the magnetic field are parallel, as in our problem, the force is zero due to the cross product property. This means the magnetic field doesn't change the speed of the charged particle, but only the direction when not parallel.
Hence, magnetic fields are crucial for altering the path of moving charges without affecting their speed, depending on the orientation.

Lorentz Force

The Lorentz force is the total force acting on a charged particle, incorporating both electric and magnetic field effects. It is expressed as: \[ \vec{F} = q \vec{E} + q (\vec{v} \times \vec{B}) \] The first part of the equation \( q \vec{E} \) represents the force from the electric field, while the second part \( q(\vec{v} \times \vec{B}) \) represents the magnetic field's effect.
This force determines how the particle moves, depending on field orientations and relative particle velocity. In the given scenario, since the electric and magnetic forces act in potentially different ways, it simplifies to only the electric field component as the magnetic force is zero due to parallel alignment.

• The Lorentz force helps calculate motion changes in charged particles effectively.
• Understanding it is key in fields such as electromagnetism and particle physics.

Motion of Charged Particles

The motion of charged particles is significantly influenced by electric and magnetic fields. When exposed to these fields, the trajectory of a particle like an electron can change.
Given an electron in our scenario, when the electric field aligns with the electron's velocity, it accelerates the electron, increasing its speed. Conversely, if opposite, it decelerates.
Magnetic fields, however, change the particle's direction rather than speed, but only if not aligned. In our case, where the magnetic field does not contribute, the electron’s speed change depends solely on the electric field.

• Charged particle motion is affected by interaction intensity and field alignment.
• Analyzing field interactions helps predict particle path and speed.

Understanding these dynamics is crucial in applications like cyclotrons and magnetic resonance devices.