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

A particle of charge $q$ and mass $m$ starts moving from the origin under the action of an electric field, $E=E_{0}$ î and $B=B_{0}$ î with a velocity, $v=v_{0} \hat{\mathbf{j}}$. The speed of the particle will becomes $\frac{\sqrt{5}}{2} v_{0}$ after a time (a) $\frac{m v_{0}}{q E}$ (b) $\frac{m v_{0}}{2 q E}$ (c) $\frac{\sqrt{3} m v_{0}}{2 q E}$ (d) $\frac{\sqrt{5} m v_{0}}{2 q E}$

A particle of charge $q$ and mass $m$ starts moving from the origin under the action of an electric field, $E=E_{0}$ î and $B=B_{0}$ î with a velocity, $v=v_{0} \hat{\mathbf{j}}$. The speed of the particle will becomes $\frac{\sqrt{5}}{2} v_{0}$ after a time
(a) $\frac{m v_{0}}{q E}$
(b) $\frac{m v_{0}}{2 q E}$
(c) $\frac{\sqrt{3} m v_{0}}{2 q E}$
(d) $\frac{\sqrt{5} m v_{0}}{2 q E}$

Key Concepts

Work Done by Electric Field

A key insight is that only the electric field does work on the charged particle since the magnetic force, being perpendicular to the particle's velocity, does not change its kinetic energy. This concept is essential for understanding how the speed of the particle changes over time, as work done by the electric field results in changes to the particle’s kinetic energy according to the work-energy theorem.

Vector Decomposition

In problems involving electromagnetic fields, vector decomposition enables the analysis of motion in independent directions. By breaking the velocity and acceleration into their component forms, one can separately consider the influences of the electric and magnetic forces, which is especially important when the directions of the fields and the initial velocity are not aligned.

Equations of Motion for Charged Particles

By applying Newton’s second law (F = ma) to the Lorentz force, one can derive the differential equations governing the particle's motion. Solving these equations yields the time evolution of the particle's velocity and position, making this concept crucial for understanding and predicting its dynamic behavior in combined electric and magnetic fields.

Lorentz Force

The Lorentz force is the fundamental force that a charged particle experiences in electromagnetic fields, given by F = q(E + v × B). This concept is key because it describes how a charged particle is accelerated by both an electric field, which does work on the particle, and a magnetic field, which influences the direction of its motion but does not change its speed.

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