At a particular instant, an electron is traveling west to...

At a particular instant, an electron is traveling west to east with a kinetic energy of $10 \mathrm{keV}$. Earth's magnetic field has a horizontal component of $1.8 \times 10^{-5} \mathrm{T}$ north and a vertical component of $5.0 \times 10^{-5} \mathrm{T}$ down. (a) What is the path of the electron? (b) What is the radius of curvature of the path?

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through a magnetic field, given by the cross product of the particle's velocity and the magnetic field vector. This force is always perpendicular to both the velocity and the magnetic field, resulting in a change in the direction of the particle's motion without a change in its speed.

Circular Motion

When a charged particle experiences a constant magnetic force that is perpendicular to its velocity, it undergoes uniform circular motion. This means that the particle travels along a curved path with a constant speed, and the magnetic force acts as the centripetal force that keeps the particle in its circular trajectory.

Radius of Curvature

The radius of curvature in the context of a charged particle moving in a magnetic field is determined by the balance between the centripetal force required for circular motion and the magnetic Lorentz force. It can be calculated using the formula r = mv/(qB), where m is the mass, v is the speed of the particle, q is the charge, and B is the magnetic field component perpendicular to the velocity.

Kinetic Energy of Charged Particle

Kinetic energy is used to determine the speed of a charged particle when its energy is known. For an electron with a given kinetic energy, one can convert that energy into speed by using the relation KE = ½mv². The calculated speed then becomes crucial in determining the radius of the curvature when the electron moves in a magnetic field.

Magnetic Field Components

In problems involving Earth's magnetic field or any non-uniform magnetic field, it is crucial to consider the vector components of the field. The horizontal and vertical components can affect the particle’s motion differently depending on the orientation of the velocity vector, leading to trajectories that might combine circular motion with other motion components, such as a potential helical path.

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