A 1.50-T magnetic field points northeast. Find the magnitude...

A 1.50-T magnetic field points northeast. Find the magnitude and direction of the force on a $+0.50-\mathrm{C}$ charge that is moving
$10 \mathrm{~m} / \mathrm{s}$ toward the (a) north; (b) east; (c) south; (d) west; (e) southwest.

Key Concepts

Vector Cross Product

The cross product operation, as used in the Lorentz force law, produces a vector that is perpendicular to the plane formed by the two input vectors. Its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them, making it crucial for calculating the force's magnitude when the velocity and magnetic field are not parallel.

Vector Nature of Motion

Velocity is inherently a vector quantity, characterized by both speed and direction. This concept is fundamental in electromagnetism because calculating the magnetic force requires knowing not only how fast a charge is moving but also in which direction it is moving relative to the magnetic field.

Magnetic Field as a Vector

A magnetic field is represented as a vector quantity, meaning it has both magnitude and direction. Understanding its orientation in space is crucial when applying the Lorentz force law, as the relative angle between the velocity of a charge and the magnetic field affects the resulting force.

Lorentz Force Law

This is the fundamental principle that describes the force experienced by a charged particle moving in electric and magnetic fields. In contexts involving magnetic fields, it is expressed as F = q(v × B), which tells us that the magnitude and direction of the force depend on the charge q, its velocity v, and the magnetic field B.

Right-Hand Rule

This rule is a mnemonic that helps determine the direction of the cross product vector. By pointing the fingers of the right hand along the direction of the first vector (velocity) and curling them towards the second vector (magnetic field), the thumb points in the direction of the resultant force, which is essential for visualizing and predicting the force’s orientation.

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