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Chapter 26: Problem 2

An electron moving with velocity \(\vec{v}\) through a magnetic field \(\vec{B}\) experiences a magnetic force \(F .\) Which of the vectors \(\vec{F}, \vec{v},\) and \(\vec{B}\) must be at right angles?

Short Answer

Expert verified
The vector \( \vec{F} \) must be at right angles with both vector \( \vec{v} \) and vector \( \vec{B} \).

Step by step solution

01

Analyze the Lorentz force law

The magnetic force acting on a moving charge in a magnetic field is given by the equation \( F = qvBsin θ \), where q is the charge, v is its velocity, B is the magnetic field, and θ is the angle between v and B.
02

Observe the relation between the force and the angle between the velocity and the magnetic field

From this equation, it can be seen that the magnetic force is maximized when the angle θ between v and B is 90 degrees (since sin 90 degrees equals 1). This means that the magnetic force is perpendicular to both the velocity of the moving charge and the magnetic field.
03

Determine which vectors must be at right angles

From the previous observation, it can be concluded that vector \( \vec{F} \), the magnetic force, must be at right angles with both vector \( \vec{v} \), the velocity of the charge, and vector \( \vec{B} \), the magnetic field. Furthermore, by the nature of the cross product, \( \vec{v} \) and \( \vec{B} \) can have any angle relative to each other, they are not specifically required to be at right angles.

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