∙A particle with mass 1.81 ×10^-3 kg and charge of +1.22 ×10^-8 Chas, at a given instant, a velo

step 1 For the given problem, we have a following situation. A magnetic field is applied in this direct

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

\bullet A particle with mass $1.81 \times 10^{-3} \mathrm{kg}$ and charge of $+1.22 \times 10^{-8} \mathrm{Chas},$ at a given instant, a velocity of $3.00 \times 10^{4} \mathrm{m} / \mathrm{s}$ along the $+y$ -axis, as shown in Figure $20.58 .$ What are the magnitude and direction of the particle's acceleration produced by a magnetic field of magnitude 1.25 $\mathrm{T}$ in the $x y-$ plane, directed at an angle of $45.0^{\circ}$ counterclockwise from the $+x$ -axis?

Key Concepts

Vector Decomposition

Vector decomposition involves breaking a vector into its component parts along specified axes. This is essential in problems with forces and fields oriented at arbitrary angles, as it allows for the analysis of the individual components of both velocity and magnetic field. By resolving vectors into perpendicular components, one can accurately determine the individual contributions to the overall force and resulting acceleration.

Charged Particle Motion in a Magnetic Field

When a charged particle moves in a magnetic field, the force it experiences (as given by the Lorentz force) causes it to accelerate, changing the direction of its motion without altering the speed if only magnetic forces are considered. This leads to circular or helical trajectories, depending on the component of the velocity that is perpendicular or parallel to the magnetic field.

Lorentz Force

The Lorentz force refers to the force exerted on a charged particle moving through electric and magnetic fields. In contexts involving only magnetic fields, it is given by F = q(v × B), where q is the charge, v is the velocity vector, and B is the magnetic field vector. This force is always perpendicular to both the velocity of the particle and the magnetic field, dictating the subsequent motion of the particle.

Vector Cross Product

The cross product of two vectors is a fundamental operation in vector calculus that results in a vector perpendicular to both of the original vectors. In the context of magnetic forces, it is used to determine both the magnitude (|q||v||B|sin?) and the direction of the force acting on a moving charge, with the angle ? representing the angle between the velocity and magnetic field vectors.

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