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When an electron enters a magnetic field at an angle, it doesn't just move in a straight line or simple circle. Instead, it follows a helical, or spiral-like, path. This happens because part of the electron's velocity is directed along the magnetic field lines, while the rest is perpendicular. The component of velocity parallel to the magnetic field lets the electron keep moving forward. Meanwhile, the perpendicular component causes it to circle around the magnetic lines.

• Parallel motion keeps the electron moving along the field.
• Perpendicular motion creates circular orbits around the field lines.

Together, these motions create a helical or spiral path. By calculating both velocity components, we can find the radius of this helical path. This helps describe how tightly the electron spirals as it moves through the field.

Electrons have a negative charge, meaning that they're inherently responsive to magnetic fields. When an electron with velocity \((32 \hat{i}+40 \hat{j}) \, \text{km/s}\) enters a magnetic field \((60 \hat{i} \, \mu \text{T})\), we can break down its motion into parts.

• The perpendicular velocity component, calculated as \(40 \, \text{km/s} = 40 \times 10^3 \, \text{m/s}\), governs the circular motion.
• The parallel velocity component, \(32 \, \text{km/s} = 32 \times 10^3 \, \text{m/s}\), ensures linear forward motion.

The magnetic field doesn't change the speed of the electron but redirects it into a spiral. By analyzing these motions, the helix's radius and pitch (or spiral step distance) can be determined.>The radius is determined by the mass, charge, and perpendicular velocity of the electron as well as the strength of the magnetic field.>The pitch is derived from the period of rotation and the parallel velocity. Understanding these aspects shows how the electron's path morphs into a helix.

Understanding the direction of electron motion in a magnetic field involves a handy tool known as the right-hand rule. Although the right-hand rule is typically applied to positively charged particles, for electrons, which are negatively charged, the rule is applied inversely.To use the right-hand rule:

• Point your thumb in the direction of the magnetic field \(\vec{B}\).
• Let your fingers follow the direction of the electron's perpendicular velocity \(\vec{v}_\perp\).
• Your palm's direction then indicates the electron's force, guiding the spiral motion.

In this specific problem, pointing your right thumb along \(\hat{i}\) and curling your fingers along \(\hat{j}\), determines that the spiral, as viewed from the entrance, moves counterclockwise. This intuitive method simplifies visualizing particle movements in a magnetic field, reinforcing the concept that the magnetic force acts perpendicular to both the velocity and field.