An electron with kinetic energy 2.5 keV moving along the positive direction of an x axis enters

step 1 In this question, we have an electron, has a kinetic energy of 2 .5 keb, it's moving along the x

An electron with kinetic energy 2.5 keV moving along the...

An electron with kinetic energy $2.5 \mathrm{keV}$ moving along the positive direction of an $x$ axis enters a region in which a uniform electric field of magnitude $10 \mathrm{kV} / \mathrm{m}$ is in the negative direction of the $y$ axis. A uniform magnetic field $\vec{B}$ is to be set up to keep the electron moving along the $x$ axis, and the direction of $\vec{B}$ is to be chosen to minimize the required magnitude of $\vec{B}$. In unit-vector notation, what $\vec{B}$ should be set up?

Key Concepts

Lorentz Force

The Lorentz force law combines both electric and magnetic forces acting on a charged particle, given by F = q (E + v × B). This principle is fundamental in predicting and controlling the motion of charged particles in electromagnetic fields, ensuring that the sum of forces determines the particle’s acceleration.

Force Cancellation for Uniform Motion

For a charged particle to continue moving with constant velocity (i.e., in uniform motion) even in the presence of electric and magnetic fields, the net force acting on it must be zero. This requires that the magnetic force exactly cancels the electric force, which is a key requirement in designing systems where the trajectory of the particle must remain unchanged.

Cross Product and Right-Hand Rule

The magnetic force on a charged particle is determined by the vector cross product v × B, which indicates both the magnitude and the direction of the force. Understanding how to apply the right?hand rule (taking into account the sign of the charge) is essential for determining the appropriate orientation of the magnetic field needed to achieve force cancellation.

Kinetic Energy and Velocity Relationship

Determining the speed of a charged particle from its kinetic energy is essential when calculating forces and magnetic field magnitudes. The relation K = ½ mv² allows one to compute the particle’s velocity, which then factors into determining the magnitude of the magnetic force through the Lorentz force equation.

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