A charged particle of mass m and charge q is projected into...

A charged particle of mass $m$ and charge $q$ is projected into a uniform magnetic field of induction $B$ with speed $v$ which is perpendicular to $B$. The width of the magnetic field is $d$. the impulse imparted to the particle by the field is $(d<<m v / q B)$ $$ \begin{array}{ccccc} q & \times & \times & \times & \times \\ \bullet^{\bullet} & \times & \times & \times \vec{B} \times \\ & \times & \times & \times & \times \end{array} $$ (A) $q B v$ (B) $m v / q B$ (C) $q B d$ (D) $2 m v^{2} / q B$

A charged particle of mass $m$ and charge $q$ is projected into a uniform magnetic field of induction $B$ with speed $v$ which is perpendicular to $B$. The width of the magnetic field is $d$. the impulse imparted to the particle by the field is $(d<<m v / q B)$
$$
\begin{array}{ccccc}
q & \times & \times & \times & \times \\
\bullet^{\bullet} & \times & \times & \times \vec{B} \times \\
& \times & \times & \times & \times
\end{array}
$$
(A) $q B v$
(B) $m v / q B$
(C) $q B d$
(D) $2 m v^{2} / q B$

Key Concepts

Lorentz Force

The Lorentz force is the fundamental force exerted on a charged particle moving in a magnetic field, expressed as F = q (v × B). It acts perpendicular to both the velocity of the particle and the direction of the magnetic field, causing the particle to undergo circular or helical motion rather than a linear acceleration.

Impulse

Impulse is defined as the product of the force applied and the time interval over which it acts, and it is equivalent to the change in momentum of the object. It provides a way to quantify the effect of forces that act over short time periods or small distances.

Time of Interaction (d/v)

When a charged particle moves through a finite region occupied by a magnetic field, the time during which the particle experiences the force is determined by dividing the width of the magnetic field (d) by the particle's speed (v). This time interval is critical for calculating the impulse, as it represents the duration over which the Lorentz force acts on the particle.

Small Deflection Approximation

In many practical situations, when the spatial extent of the magnetic field is much smaller than the characteristic length scale of the particle's curved path (i.e., d is much less than the radius of curvature), the deflection of the particle is minimal. This allows for simplifying assumptions, such as treating the particle’s speed as approximately constant during its passage through the field, and using a linear approximation to compute the impulse.

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