91Ó°ÊÓ

The Lorentz force is fundamental in understanding how charged particles interact with magnetic fields. It is expressed as \( \vec{F} = q(\vec{v} \times \vec{B}) \). Here, \( q \) represents the charge of the particle, \( \vec{v} \) is the velocity at which it travels, and \( \vec{B} \) stands for the magnetic field vector. This equation tells us that the force on the particle not only depends on these variables but is also the result of a cross product, which means it will always be perpendicular to both \( \vec{v} \) and \( \vec{B} \). This perpendicular nature is why charged particles perform spiral or circular motions in magnetic fields.

Calculating the cross product is key in determining the Lorentz force. The cross product \( \vec{v} \times \vec{B} \) is calculated using a determinant:+ When you set up the determinant with the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and fill in the velocity \( (2.0 \times 10^6 \hat{i} + 3.0 \times 10^6 \hat{j}) \) and the magnetic field \( (0.030 \hat{i} - 0.15 \hat{j}) \), you get specific components.+ This results in the vector \( (0 \hat{i} + 0 \hat{j} - 3.9 \times 10^5 \hat{k}) \), which signifies the direction of the magnetic force relative to the motion and field.Calculating this correctly is crucial because it tells us in which direction the velocity of the particle will alter as it travels through the magnetic field.

The charge of an electron is \( -1.6 \times 10^{-19} \) C, while a proton carries a charge of \( 1.6 \times 10^{-19} \) C. Although they have opposite signs, the magnitude of their charges is the same, which means that they will experience equal magnitude of force in a given magnetic field if their velocities are identical. However, due to the negative charge, the force on the electron will be in the opposite direction compared to the proton. Understanding these charges' effects helps predict the behaviors of electron and proton in the same magnetic environment, such as their paths diverging based on their charge type when subjected to the same forces.

A magnetic field is a vector field that exerts force on particles that are in motion. It is represented as \( \vec{B} \) and measured in teslas (T). The direction of the field affects the direction of the force applied to a moving charge. In the given case, the field is expressed by \( (0.030 \hat{i} - 0.15 \hat{j}) \) T, showing it has components along \( \hat{i} \) and \( \hat{j} \) axes. The minus sign indicates the field's direction along the \( \hat{j} \)-axis is opposite to the conventional positive direction. When a charged particle such as an electron or proton moves through this field, the magnetic force calculated takes these vector components into account, determining the eventual path and behavior of the particle.