A proton traveling at 3.60 km / ssuddenly enters a uniform magnetic field of 0.750 T, travelin

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A proton traveling at 3.60 km / ssuddenly enters a uniform...

A proton traveling at $3.60 \mathrm{~km} / \mathrm{s}$suddenly enters a uniform magnetic field of $0.750 \mathrm{~T},$ traveling at an angle of $55.0^{\circ}$ with the field lines (Figure 20.55 ).(a) Find the magnitude and direction of the force this magnetic field exerts on the proton.(b) If you can vary the direction of the proton's velocity, find the magnitude of the maximum and minimum forces you could achieve, and show how the velocity should be oriented to achieve these forces. (c) What would the answers to part (a) be if the proton were replaced by an electron traveling in the same way as the proton?

Key Concepts

Magnetic Force on a Moving Charged Particle

This concept involves understanding that a charged particle moving in a magnetic field experiences a force given by the equation F = q(v × B). The magnitude of this force depends on the particle's charge, its speed, the strength of the magnetic field, and the sine of the angle between the velocity and the magnetic field. The force always acts perpendicular to both the velocity of the particle and the direction of the magnetic field.

Vector Cross Product and Sine Dependency

The calculation of the magnetic force makes use of the vector cross product, which mathematically captures how two perpendicular components interact. In this context, the magnitude of the force is determined by the product of the charge, the speed, the magnetic field strength, and the sine of the angle between the velocity and the magnetic field vectors. This dependence on the sine of the angle means that the force is maximum when the velocity vector is perpendicular to the magnetic field and zero when they are parallel.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic force on a positive charge moving in a magnetic field. By pointing the fingers of the right hand in the direction of the velocity and then curling them towards the magnetic field vector, the outstretched thumb indicates the direction of the force. This rule helps to visually and intuitively understand the three-dimensional orientation of the force.

Maximum and Minimum Magnetic Force Conditions

The magnitude of the magnetic force varies with the sine of the angle between the velocity and the magnetic field. The maximum force is achieved when this angle is 90° (resulting in sin 90° = 1), meaning the velocity is perpendicular to the magnetic field. Conversely, the minimum force occurs when the velocity is parallel to the magnetic field (0° or 180°), because sin 0° = 0, resulting in zero magnetic force.

Charge Sign Effects

The sign of the charge of the particle affects the direction of the magnetic force. For positive charges, the direction of the force is given directly by the right-hand rule. For negative charges, such as electrons, the force direction is opposite to that given by the right-hand rule. This reversal is crucial for understanding the behavior of different charged particles in a magnetic field.

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