A charged particle of mass m and charge q is accelerated...

A charged particle of mass $m$ and charge $q$ is accelerated through a potential difference of $V$ volt. It enters a region of uniform magnetic field which is directed perpendicular to the direction of motion of the particie. The particle will move on a circular path of radius given by: (a) $\sqrt{\left(\frac{V m}{q B^{2}}\right)}$ (b) $\frac{2 V m}{q B^{2}}$ (c) $\sqrt{\left(\frac{2 V_{m}}{q}\right)}\left(\frac{1}{B}\right)$ (d) $\sqrt{\left(\frac{V m}{q}\right)}\left(\frac{1}{B}\right)$

A charged particle of mass $m$ and charge $q$ is accelerated through a potential difference of $V$ volt. It enters a region of uniform magnetic field which is directed perpendicular to the direction of motion of the particie. The particle will move on a circular path of radius given by:
(a) $\sqrt{\left(\frac{V m}{q B^{2}}\right)}$
(b) $\frac{2 V m}{q B^{2}}$
(c) $\sqrt{\left(\frac{2 V_{m}}{q}\right)}\left(\frac{1}{B}\right)$
(d) $\sqrt{\left(\frac{V m}{q}\right)}\left(\frac{1}{B}\right)$

Key Concepts

Magnetic Lorentz Force

A charged particle moving in a magnetic field experiences a force called the Lorentz force, which is given by F = qvB when the velocity is perpendicular to the magnetic field. This force is responsible for the curvature of the particle's path in the field.

Centripetal Force and Circular Motion

When the Lorentz force acts as the centripetal force, it causes the particle to move in a circular path. The centripetal force needed to keep the particle in circular motion is mv²/r, setting this equal to the magnetic force qvB allows one to derive the expression for the radius of the circular path.

Energy Conservation

When a charged particle is accelerated through a potential difference, its electric potential energy is converted into kinetic energy. This conversion is described by the equation ½mv² = qV, which provides the magnitude of the particle’s velocity after acceleration.

Radius of Circular Motion

The radius of the circular path for a charged particle in a magnetic field can be determined by combining the expressions for kinetic energy gained from acceleration and the centripetal force requirement. This leads to the result r = (mv)/(qB), which, with the appropriate substitution for v from the energy conservation equation, gives the expression for the radius in terms of the accelerating potential, mass, charge, and magnetic field.

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