An electron that has an instantaneous velocity of v⃗=(2.0 ×10^6 m / s) î+(3.0 ×10^6 m / s) ĵ i

step 1 All right, So in this problem, we are given that we have an electron with some velocity and it i

An electron that has an instantaneous velocity of v⃗=(2.0...

An electron that has an instantaneous velocity of $$ \vec{v}=\left(2.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{i}}+\left(3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{j}} $$ is moving through the uniform magnetic field $\vec{B}=(0.030 \mathrm{~T}) \hat{\mathrm{i}}-$ $(0.15 \mathrm{~T}) \hat{\mathrm{j}} .$ (a) Find the force on the electron due to the magnetic field. (b) Repeat your calculation for a proton having the same velocity.

An electron that has an instantaneous velocity of
$$
\vec{v}=\left(2.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{i}}+\left(3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{j}}
$$
is moving through the uniform magnetic field $\vec{B}=(0.030 \mathrm{~T}) \hat{\mathrm{i}}-$ $(0.15 \mathrm{~T}) \hat{\mathrm{j}} .$ (a) Find the force on the electron due to the magnetic field. (b) Repeat your calculation for a proton having the same velocity.

Key Concepts

Uniform Magnetic Field

A uniform magnetic field is one that has a constant magnitude and direction at every point in space. This uniformity simplifies the analysis of charged particle dynamics because the force experienced by a charged particle is consistent across its trajectory, assuming no other forces act on it. It is an important concept for understanding magnetic interactions in controlled environments, such as in laboratory experiments or certain practical applications in physics and engineering.

Charge Sign and Its Effects

The sign of the charge of a particle plays a critical role in determining the direction of the magnetic force as dictated by the right-hand rule. For a positive charge, the force direction follows the standard right-hand rule applied to v and B. For a negative charge, the force direction is opposite to that predicted by the right-hand rule. This concept is essential when comparing the behavior of particles like electrons (negative charge) and protons (positive charge) in a magnetic field.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through electric and magnetic fields. In the absence of an electric field, the force is given by F = q(v × B), where q is the charge of the particle, v is its velocity, and B is the magnetic field. This concept is central to understanding how charged particles behave in magnetic environments, as it governs the magnitude and direction of the force acting on them.

Vector Cross Product

The vector cross product is a mathematical operation used to determine a vector that is perpendicular to two given vectors. In the context of the Lorentz force, the cross product of the velocity vector and the magnetic field vector determines both the magnitude and the direction of the force on the charged particle. Understanding the cross product is crucial, as it explains why the force is maximized when the velocity and magnetic field are perpendicular and zero when they are parallel.

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