A particle of mass m and charge q is released from rest at...

A particle of mass $m$ and charge $q$ is released from rest at the origin as shown in the figure. The speed of the particle when it has travelled a distance $d$ along the $z$ -axis is given by: (a) $\sqrt{\left(\frac{2 q E_{0} d}{m}\right)}$ (b) $\sqrt{\left(\frac{E_{0} d}{B_{0}}\right)}$ (c) $\sqrt{\left(\frac{m}{q E_{0} d}\right)}$ (d) $\sqrt{\left(\frac{2 m}{q E_{0} d}\right)}$

A particle of mass $m$ and charge $q$ is released from rest at the origin as shown in the figure. The speed of the particle when it has travelled a distance $d$ along the $z$ -axis is given by:
(a) $\sqrt{\left(\frac{2 q E_{0} d}{m}\right)}$
(b) $\sqrt{\left(\frac{E_{0} d}{B_{0}}\right)}$
(c) $\sqrt{\left(\frac{m}{q E_{0} d}\right)}$
(d) $\sqrt{\left(\frac{2 m}{q E_{0} d}\right)}$

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Key Concepts

Electric Force on a Charged Particle

When a charged particle is placed in an electric field, it experiences an electric force given by F = qE. This force acts in the direction of the electric field if the charge is positive, or opposite if the charge is negative. This concept is fundamental for analyzing the motion of charged particles under electric fields since the force directly influences the particle's acceleration.

Uniform Acceleration Kinematics

Once the force on a charged particle is known, the resulting acceleration can be found using Newton's second law (a = F/m). With a constant acceleration, the equations of motion (or kinematic equations) can be applied. In this context, the relation v^2 = v0^2 + 2ad is particularly useful, enabling calculation of the particle’s speed after traveling a certain distance from rest.

Work-Energy Principle

The work done on the charged particle by the electric force results in a change in its kinetic energy. By equating the work (which is the product of force and displacement, qEd) to the increase in kinetic energy (½mv²), one can derive an expression for the speed of the particle. This principle provides an alternative approach to solving problems involving the acceleration of particles in electric fields.

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