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Step-by-Step Explanation
1. Understanding the Forces on the Charged Particle
A charged particle in an electric field $\vec{E}$ and magnetic field $\vec{B}$ experiences the Lorentz force given by:
$$ \vec{F} = q \Bigl(\vec{E} + \vec{v} \times \vec{B}\Bigr) $$
where
$q$ is the charge on the particle,
$\vec{v}$ is the velocity of the particle,
$\vec{E}$ is the electric field, and
$\vec{B}$ is the magnetic field.
2. Initial Condition: Particle Released from Rest
The particle starts from rest, so initially its velocity $\vec{v}_0 = 0$. This means at the instant it is released, the magnetic part of the force $q(\vec{v} \times \vec{B})$ is zero because $\vec{v} = 0$ initially.
3. Parallel Electric and Magnetic Fields
It is given that $\vec{E}$ and $\vec{B}$ are parallel to each other. As the particle starts moving under the influence of the electric field, its velocity will be directed along $\vec{E}$. Because $\vec{E}$ and $\vec{B}$ are parallel, the velocity $\vec{v}$ will also remain parallel (or antiparallel) to $\vec{B}$.
4. Magnetic Force When Velocity and Magnetic Field are Parallel
The magnetic force term $q(\vec{v} \times \vec{B})$ depends on the cross product of $\vec{v}$ and $\vec{B}$. If $\vec{v}$ is parallel (or antiparallel) to $\vec{B}$, then $\vec{v} \times \vec{B} = 0$. Hence, there is no magnetic force acting on the particle once it starts moving along $\vec{E}$.
5. Net Force and Resulting Path
The only net force on the charged particle comes from the electric field, which acts in a fixed direction. Consequently, the particle accelerates in a straight line along the direction of the electric field.
6. Conclusion
Because there is no bending force (no magnetic force component) and the electric force is constant in direction, the particle moves in a straight line.
