A particle carrying a -1.4-μC charge moves at 185 m / s through a uniform 45-mT magnetic field

step 1 This question belongs to the magnetic force in which we have a particle of charge Q equals to mi

A particle carrying a -1.4-μC charge moves at 185 m / s...

A particle carrying a $-1.4-\mu$ C charge moves at $185 \mathrm{~m} / \mathrm{s}$ through a uniform $45-\mathrm{mT}$ magnetic field pointing in the $+y$ -direction. The resulting force on the particle is $9.3 \mu \mathrm{N}$ in the $+z$ -direction. In what direction is the particle moving?

Key Concepts

Coordinate System and Vector Directions

Understanding vector directions in a defined coordinate system is essential when solving problems involving cross products. In these scenarios, each axis (typically x, y, and z) represents a distinct directional component, and the orientation of the velocity, magnetic field, and force vectors must be appropriately assigned and interpreted with respect to these axes.

Effect of Charge Sign

The sign of the charge plays a critical role in determining the direction of the magnetic force. While the right-hand rule provides the direction for a positively charged particle, a negative charge reverses this direction because the force becomes antiparallel to the right-hand rule result. This inversion is key when analyzing the movement of negatively charged particles in a magnetic field.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the resultant vector in a cross product. When applied to F = q(v × B), it shows that if you point the fingers of your right hand in the direction of v and then rotate them toward B, your thumb will point in the direction of the force for a positive charge. For a negative charge, the force direction is opposite to that indicated by the right-hand rule.

Lorentz Force Law

This law describes the force exerted on a charged particle moving in a magnetic field. It is expressed as F = q(v × B), where F is the force, q is the charge, v is the velocity, and B is the magnetic field. The law highlights that the force is perpendicular to both the velocity of the particle and the magnetic field, and its magnitude depends on the charge value and the sine of the angle between v and B.

Cross Product

The cross product of two vectors, such as velocity and magnetic field, determines a new vector that is perpendicular to both original vectors. The magnitude of the cross product is given by |v||B|sin(?), where ? is the angle between v and B, and its direction is determined by the right-hand rule.

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