Protons having a kinetic energy of 5.00 MeV (1 eV=1.60 ×10^-19 J) are moving in the positive x d

step 1 As kinetic energy of proton is given, so first we will calculate speed of proton as we have kine

Protons having a kinetic energy of 5.00 MeV (1 eV=1.60...

Protons having a kinetic energy of 5.00 MeV $\left(1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{J}\right)$ are moving in the positive $x$ direction and enter a magnetic field $\overrightarrow{\mathbf{B}}=$ $0.0500 \hat{\mathbf{k}} \mathrm{T}$ directed out of the plane of the page and extending from $x=0$ to $x=1.00 \mathrm{m}$ as shown in Figure $\mathrm{P} 22.75 .$ (a) Ignoring relativistic effects, find the angle $\alpha$ between the initial velocity vector of the proton beam and the velocity vector after the beam emerges from the field. (b) Calculate the $y$ component of the protons' momenta as they leave the magnetic field.

Key Concepts

Momentum and Its Components

Momentum, defined as the product of mass and velocity, is a vector quantity that reflects both the magnitude and direction of a particle’s motion. When a particle's path is altered by a magnetic field, its momentum vector changes direction. Analyzing the components of the momentum vector, such as its y component, is essential in quantifying the effect of the magnetic deflection.

Kinetic Energy and Velocity Relation

The kinetic energy of a particle is directly related to its speed through the relation KE = 1/2 mv². This relationship allows one to determine the velocity of the particle when its kinetic energy and mass are known, which is crucial for understanding the dynamics of the motion in a magnetic field.

Cyclotron Radius

The radius of the circular path followed by a charged particle in a magnetic field, often called the cyclotron radius, is given by r = mv/(qB), where m is the mass of the particle, v is its speed, q is its charge, and B is the magnetic field strength. This formula is key to predicting how much the particle will curve during its motion.

Lorentz Force

The Lorentz force is the fundamental force acting on a charged particle moving in a magnetic field. It is given by the cross product of the velocity of the particle and the magnetic field, multiplied by the charge. This force always acts perpendicular to the particle's velocity, causing the path of the particle to curve without changing its speed.

Circular Motion in a Magnetic Field

When a charged particle moves through a magnetic field, the perpendicular Lorentz force causes the particle to undergo uniform circular motion. The magnetic force acts as the centripetal force needed to keep the particle in circular motion, leading to a well-defined radius and period for its path.

Transcript

Please give Ace some feedback