In a certain region of space, there is a uniform electric field 𝐄=2.0 ×10^4 V / m to the east a

In a certain region of space, there is a uniform electric...

In a certain region of space, there is a uniform electric field $\overrightarrow{\mathbf{E}}=2.0 \times 10^{4} \mathrm{~V} / \mathrm{m}$ to the east and a uniform magnetic field $\overrightarrow{\mathbf{B}}=0.0050 \mathrm{~T}$ to the west. (a) What is the electromagnetic force on an electron moving north at $1.0 \times 10^{7} \mathrm{~m} / \mathrm{s} ?$ (b) With the electric and magnetic fields as specified, is there some velocity such that the net electromagnetic force on the electron would be zero? If so, give the magnitude and direction of that velocity. If not, explain briefly why not.

Key Concepts

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in the presence of electric and magnetic fields. It is given by the sum of the electric force, which is the product of the charge and the electric field, and the magnetic force, which is the charge multiplied by the cross product of the particle’s velocity and the magnetic field. This concept is fundamental in understanding how charged particles behave under electromagnetic influences.

Electric Field

An electric field represents a region of space around a charged particle or distribution of charge within which other charged objects experience a force. The field is defined such that the force on a test charge is proportional to the magnitude of the charge and the electric field, acting in the direction of the field for positive charges and opposite for negative charges.

Magnetic Field

A magnetic field is produced by moving electric charges (currents) and magnetic dipoles, influencing other moving charges or magnetic materials. The magnetic force experienced by a moving charge is dependent on the velocity of the charge and the magnetic field strength, and it acts perpendicular to both the velocity and the magnetic field vector, a phenomenon explained by the right-hand rule.

Vector Cross Product

The cross product is a mathematical operation on two vectors that results in a vector perpendicular to both original vectors. In the context of electromagnetic forces, it is used to determine the direction of the magnetic force on a moving charge, based on its velocity and the magnetic field. The magnitude of the resulting vector describes the strength of the force, following the sine of the angle between the input vectors.

Force Cancellation Condition

Force cancellation refers to the scenario in which two forces equal in magnitude but opposite in direction act on a charged particle, resulting in a net force of zero. In electromagnetic contexts, this often involves adjusting the velocity of the charge so that the magnetic force precisely offsets the electric force, a condition that is critical in many applications such as velocity selectors and particle beam dynamics.