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A charged particle enters a uniform magnetic field with velocity vector at an angle of 45° with the magnetic field. The pitch of the helical path followed by the particle is p. The radius of the helix is given by p/nπ. The value of n is
Correct answer is '2'. Can you explain this answer?
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A charged particle enters a uniform magnetic field with velocity vecto...
The correct answer is: 2
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Understanding the Helical Motion of a Charged Particle
When a charged particle enters a uniform magnetic field at an angle, it experiences a force perpendicular to both its velocity and the magnetic field, resulting in a helical path.
Components of Velocity
- The velocity vector can be split into two components:
- Parallel Component (v_parallel): This component is aligned with the magnetic field.
- Perpendicular Component (v_perpendicular): This component is perpendicular to the magnetic field.
Given that the angle is 45 degrees, both components are equal:
- v_parallel = v * cos(45°)
- v_perpendicular = v * sin(45°)
Circular Motion
- The perpendicular component of the velocity causes circular motion:
- The radius (r) of the circular path is determined by the equation: r = mv_perpendicular / (qB), where m is mass, q is charge, and B is the magnetic field strength.
Pitch of the Helix
- The pitch (p) is the distance the particle moves parallel to the magnetic field in one complete revolution:
- p = v_parallel * T, where T is the time period of one complete circular motion.
Finding the Radius
- Using the relation for T (the time period), we find:
- T = (2π * r) / v_perpendicular.
- Substituting the expressions, we can derive that the helical path's characteristics relate to the pitch and radius through specific ratios.
Conclusion: Value of n
- The radius of the helix is given as r = p / (nπ).
- Upon solving the relations, it becomes evident that n takes the value of 2, showcasing that the helical path's geometry is consistent with the proportions of the velocity components and the effects of the magnetic field.
This leads us to conclude that the correct answer for n in this scenario is indeed 2.
When a charged particle enters a uniform magnetic field at an angle, it experiences a force perpendicular to both its velocity and the magnetic field, resulting in a helical path.
Components of Velocity
- The velocity vector can be split into two components:
- Parallel Component (v_parallel): This component is aligned with the magnetic field.
- Perpendicular Component (v_perpendicular): This component is perpendicular to the magnetic field.
Given that the angle is 45 degrees, both components are equal:
- v_parallel = v * cos(45°)
- v_perpendicular = v * sin(45°)
Circular Motion
- The perpendicular component of the velocity causes circular motion:
- The radius (r) of the circular path is determined by the equation: r = mv_perpendicular / (qB), where m is mass, q is charge, and B is the magnetic field strength.
Pitch of the Helix
- The pitch (p) is the distance the particle moves parallel to the magnetic field in one complete revolution:
- p = v_parallel * T, where T is the time period of one complete circular motion.
Finding the Radius
- Using the relation for T (the time period), we find:
- T = (2π * r) / v_perpendicular.
- Substituting the expressions, we can derive that the helical path's characteristics relate to the pitch and radius through specific ratios.
Conclusion: Value of n
- The radius of the helix is given as r = p / (nπ).
- Upon solving the relations, it becomes evident that n takes the value of 2, showcasing that the helical path's geometry is consistent with the proportions of the velocity components and the effects of the magnetic field.
This leads us to conclude that the correct answer for n in this scenario is indeed 2.
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A charged particle enters a uniform magnetic field with velocity vector at an angle of 45° with the magnetic field. The pitch of the helical path followed by the particle isp. The radius of the helix is given by p/nπ.The value ofnisCorrect answer is '2'. Can you explain this answer?
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