An electron experiences the greatest force as it travels 2.8...

An electron experiences the greatest force as it travels $2.8 \times 10^{6} \mathrm{~m} / \mathrm{s}$ in a magnetic field when it is moving northward. The force is vertically upward and of magnitude $8.2 \times 10^{-13} \mathrm{~N}$. What is the magnitude and direction of the magnetic field?

Key Concepts

Lorentz Force

The Lorentz Force is the force experienced by a charged particle moving in electric and magnetic fields. In the context of magnetic fields, it is given by the equation F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. The magnitude of this force is |F| = |q|vB sin(?), with ? being the angle between the velocity and the magnetic field vectors. This law is fundamental in analyzing the motion of charged particles in magnetic fields.

Magnetic Force on a Moving Charge

The magnetic force on a moving charge arises from the interaction between the charge’s velocity and the surrounding magnetic field. For maximum force, the motion is perpendicular to the magnetic field (? = 90°). This force not only causes the particle to change direction but also helps in calculating the magnitude of the magnetic field when the force and velocity are known. This concept is widely applied in devices like mass spectrometers and cyclotrons.

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 the direction of the force. The magnitude of the cross product incorporates the sine of the angle between the two vectors, reinforcing why the force is maximized when the two vectors are perpendicular.

Right-hand and Left-hand Rules

The right-hand rule is a mnemonic used to determine the direction of the force exerted on a positive charge moving in a magnetic field by taking the cross product of the velocity and magnetic field vectors. However, for negatively charged particles such as electrons, the direction of the force is opposite to that given by the right-hand rule. This distinction is crucial when analyzing motion in magnetic fields, as it requires one to reverse the expected direction determined by the standard rule.

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