A particle with mass 1.81 ×10^-3 kg and a charge of 1.22 ×10^-8 C has, at a given instant, a vel

step 1 From inter second la, we know that F is equal to MA as well as we know that the force on charge

A particle with mass 1.81 ×10^-3 kg and a charge of 1.22...

A particle with mass $1.81 \times 10^{-3} \mathrm{kg}$ and a charge of $1.22 \times 10^{-8} \mathrm{C}$ has, at a given instant, a velocity $\vec{\boldsymbol{v}}=(3.00 \times$ $10^{4} \mathrm{m} / \mathrm{s} ) \hat{\boldsymbol{J}}$ . What are the magnitude and direction of the particle's acceleration produced by a uniform magnetic field $\vec{\boldsymbol{B}}=(1.63 \mathrm{T}) \hat{\boldsymbol{\imath}}+$ $(0.980 \mathrm{T}) \hat{\boldsymbol{l}}$ ?

Key Concepts

Lorentz Force

The Lorentz force describes the force on a charged particle moving in an electromagnetic field. It is given by F = q(E + v × B), where q is the charge, v is the velocity, and B is the magnetic field. In cases with only a magnetic field present, the force simplifies to F = q(v × B), acting perpendicular to both the velocity and the field.

Cross Product in Vector Calculus

The cross product is an operation on two vectors that produces a third vector perpendicular to the plane formed by the initial vectors. Its magnitude is determined by the product of the magnitudes of the vectors and the sine of the angle between them, while its direction is given by the right-hand rule. This concept is essential for calculating forces in electromagnetic problems where directions of resulting vectors are important.

Uniform Magnetic Field

A uniform magnetic field is one where both its magnitude and direction are constant throughout space. This consistency allows for straightforward analysis of the motion of charged particles since the force experienced (due to the magnetic field) remains uniformly defined, leading to predictable, often circular, paths.

Circular Motion of Charged Particles

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic component of the Lorentz force causes it to move in a circular path. The force acts as a centripetal force, continuously changing the direction of the velocity vector without affecting its magnitude, which results in uniform circular motion.

Acceleration due to Magnetic Force

Acceleration in electromagnetic contexts is deduced by applying Newton's second law (a = F/m). For a charged particle experiencing a magnetic force, the acceleration is determined by dividing the magnetic force (q(v × B)) by the particle's mass. This acceleration is always perpendicular to the velocity, resulting in a change in direction rather than in speed.

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