At one instant, v⃗=(-2.00 î+4.00 ĵ-6.00 k̂) m / s is the ve- locity of a proton in a uniform mag

step 1 In this question, we are given the motion of proton in the region of uniform magnetic field. So

At one instant, v⃗=(-2.00 î+4.00 ĵ-6.00 k̂) m / s is the ve-...

At one instant, $\vec{v}=(-2.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}}-6.00 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$ is the ve- locity of a proton in a uniform magnetic field $\vec{B}=(2.00 \hat{\mathrm{i}}-$ $4.00 \mathrm{j}+8.00 \mathrm{k}) \mathrm{mT}$. At that instant, what are (a) the magnetic force $\vec{F}$ acting on the proton, in unit-vector notation, (b) the angle between $\vec{v}$ and $\vec{F}$, and $(\mathrm{c})$ the angle between $\vec{v}$ and $\vec{B}$ ?

At one instant, $\vec{v}=(-2.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}}-6.00 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$ is the ve-
locity of a proton in a uniform magnetic field $\vec{B}=(2.00 \hat{\mathrm{i}}-$ $4.00 \mathrm{j}+8.00 \mathrm{k}) \mathrm{mT}$. At that instant, what are (a) the magnetic force $\vec{F}$ acting on the proton, in unit-vector notation, (b) the angle between $\vec{v}$ and $\vec{F}$, and $(\mathrm{c})$ the angle between $\vec{v}$ and $\vec{B}$ ?

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Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in electric and magnetic fields. In the case of a pure magnetic field, the force is given by the cross product of the particle's velocity vector and the magnetic field vector, scaled by the particle's charge.

Cross Product

The cross product of two vectors is an operation that results in a new vector perpendicular to the plane formed by the original vectors. Its magnitude is proportional to the sine of the angle between the two vectors and is crucial in determining both the magnitude and direction of the magnetic force.

Angle Between Vectors

The angle between vectors is a key concept that helps in understanding the geometric relationships in vector operations. Specifically, for the magnetic force, knowing the angle between the velocity and the magnetic field (and consequently the force) is essential because the force is always perpendicular to the velocity vector.

Uniform Magnetic Field

A uniform magnetic field is one that has a constant magnitude and direction at every point. This uniformity simplifies the analysis of the motion of charged particles, as the magnetic force on the particle depends only on its instantaneous velocity and the constant field.

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