An electron moves through a uniform magnetic field given by B⃗=Bx î+(3.0 Bx) ĵ. At a particular

step 1 In this question, we are given the following information. So we have electron that is moving wit

An electron moves through a uniform magnetic field given by...

An electron moves through a uniform magnetic field given by $\vec{B}=B_{x} \hat{i}+\left(3.0 B_{x}\right) \hat{j}$. At a particular instant, the electron has velocity $\vec{v}=(2.0 \hat{\mathrm{i}}+4.0 \mathrm{j}) \mathrm{m} / \mathrm{s}$ and the magnetic force acting on it is $\left(6.4 \times 10^{-19} \mathrm{~N}\right) \hat{\mathrm{k}}$. Find $B_{x}$

Key Concepts

Lorentz Force Law

The Lorentz force law describes the force on a charged particle moving in electric and magnetic fields. It is given by F = q(v × B), where F is the force, q is the charge, v is the velocity vector, and B is the magnetic field vector. This law is fundamental in electromagnetism, as it explains phenomena such as the circular motion of charged particles and the operation of devices like cyclotrons and mass spectrometers.

Vector Cross Product

The cross product is a mathematical operation on two vectors in three-dimensional space that results in a vector perpendicular to both original vectors. Its magnitude is determined by the product of the magnitudes of the two vectors and the sine of the angle between them. In the context of the Lorentz force, the cross product v × B determines the direction and magnitude of the force acting on the charged particle, making the understanding of this operation crucial.

Magnetic Field Components

A magnetic field is often represented as a vector with different components along the coordinate axes. Analyzing the components of the magnetic field allows one to understand and calculate the resultant force on a moving charge. When given specific component relations, one can use these in conjunction with the velocity vector components to determine the force via the cross product.

Sign of the Charge

The sign of the charge is important when applying the Lorentz force law because it determines the direction of the force. For example, electrons have a negative charge, which means the direction of the force experienced by an electron in a magnetic field is opposite to that predicted by the right-hand rule when applied without accounting for the sign. This is crucial for correctly determining the dynamics of the electron in the field.

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