[T] The force vector F acting on a proton with anelectric...

$[T]$ The force vector $F$ acting on a proton with anelectric charge of $1.6 \times 10^{-19} \mathrm{C}$ moving in a magnetic field $B$ where the velocity vector $V$ is given by$\mathbf{F}=1.6 \times 10^{-19}(\mathbf{v} \times \mathbf{B})$ (here, v is expressed in meters per second, $B$ in $T$, and $F$ in $N$). If the magnitude of force $F$ acting on a proton is$5.9 \times 10^{-17} \mathrm{~N}$ and the proton is moving at the speed of 300 m/sec in magnetic field $B$ of magnitude $2.4 T$, find the angle between velocity vector $V$ of the proton and magnetic field $B$. Express the answer in degrees rounded to the nearest integer.

$[T]$ The force vector $F$ acting on a proton with anelectric charge of $1.6 \times 10^{-19} \mathrm{C}$ moving in a magnetic
field $B$ where the velocity vector $V$ is given by$\mathbf{F}=1.6 \times 10^{-19}(\mathbf{v} \times \mathbf{B})$ (here, v is expressed in meters
per second, $B$ in $T$, and $F$ in $N$). If the magnitude of force $F$ acting on a proton is$5.9 \times 10^{-17} \mathrm{~N}$ and the
proton is moving at the speed of 300 m/sec in magnetic field $B$ of magnitude $2.4 T$, find the angle between velocity
vector $V$ of the proton and magnetic field $B$. Express the answer in degrees rounded to the nearest integer.

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Key Concepts

Cross Product in Vector Operations

The cross product of two vectors produces a third vector that is perpendicular to the plane formed by the original vectors. Its magnitude is given by |v × B| = |v| |B| sin(?), where ? is the angle between the vectors. This relationship is crucial for determining the component of the force perpendicular to both v and B.

Trigonometric Relation in Force Calculations

By relating the magnitude of the force to the sine of the angle between the velocity and the magnetic field, one can determine the angle if the force, charge, speed, and magnetic field magnitude are known. This involves rearranging the equation |F| = q |v| |B| sin(?) to solve for ? using the inverse sine function.

Magnetic Lorentz Force

This concept describes the force experienced by a moving charged particle in a magnetic field. The force is given by the vector equation F = q(v × B), where q is the charge, v is the velocity vector, B is the magnetic field vector, and × represents the cross product. This force is always perpendicular to both the velocity of the particle and the magnetic field direction.

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