A particle of charge q and mass m starts moving from origin...

A particle of charge $q$ and mass $m$ starts moving from origin under the action of an electric field $\vec{E}=E_{0} \hat{i}$ and magnetic field $\vec{B}=B_{0} \hat{k}$. Its velocity at $(x, 0,0)$ is $6 \hat{i}+8 \hat{j}$. The value of $x$ is (A) $\frac{25 m}{q E_{0}}$ (B) $\frac{100 m}{q B_{0}}$ (C) $\frac{50 m}{q E_{0}}$ (D) $\frac{14 m}{q E_{0}}$

A particle of charge $q$ and mass $m$ starts moving from origin under the action of an electric field $\vec{E}=E_{0} \hat{i}$ and magnetic field $\vec{B}=B_{0} \hat{k}$. Its velocity at $(x, 0,0)$ is $6 \hat{i}+8 \hat{j}$. The value of $x$ is
(A) $\frac{25 m}{q E_{0}}$
(B) $\frac{100 m}{q B_{0}}$
(C) $\frac{50 m}{q E_{0}}$
(D) $\frac{14 m}{q E_{0}}$

Key Concepts

Lorentz Force Law

This principle states that a charged particle experiences a force in the presence of electric and magnetic fields, given by F = q(E + v × B). It is fundamental in predicting and analyzing the motion of charged particles in electromagnetic fields, regardless of whether the fields are static or time-dependent.

Magnetic Forces and Work

A crucial property of magnetic forces is that they always act perpendicular to the velocity of a particle, which means they do no work on the particle. As a result, while magnetic forces can change the direction of the velocity, they do not alter the particle’s kinetic energy.

Work-Energy Theorem in Electromagnetic Contexts

In systems where charged particles are acted upon by both electric and magnetic fields, the work done by the electric field is solely responsible for the change in kinetic energy. This relationship is expressed as the work done (qE0x) equaling the kinetic energy gained, which is central to solving problems involving energy transfer under electromagnetic forces.

Motion in Uniform Electric Fields

In a uniform electric field, the acceleration of a charged particle is constant since the force on the particle is constant. This leads to energy changes that can be calculated simply by considering the work done over a displacement in the direction of the field, making it a key concept in analyzing trajectories and energy acquisition in such scenarios.

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