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Chapter 27: Problem 65

A charged particle enters into a uniform magnetic field \(B\) with velocity \(\vec{v}\) at an angle \(\theta\) as shown in figure. Then the ratio of radius to pitch of helix is: (a) \(\frac{2 \pi}{\tan \theta}\) (b) \(\tan \theta\) (c) \(\cot \theta\) (d) \(\frac{\tan \theta}{2 \pi}\)

Short Answer

Expert verified
(d) \( \frac{\tan \theta}{2 \pi} \).

Step by step solution

01

Understand the Motion of the Particle

When a charged particle moves through a magnetic field at an angle \( \theta \), its velocity can be split into two components: one parallel to the magnetic field \( v_{\parallel} = v \cdot \cos \theta \) and one perpendicular to the magnetic field \( v_{\perp} = v \cdot \sin \theta \). The perpendicular component causes the particle to move in a circular path, while the parallel component leads to linear motion along the field lines.
02

Determine the Radius of Helical Path

The radius \( r \) of the circular path can be determined using the formula: \( r = \frac{m \cdot v_{\perp}}{q \cdot B} \), where \( m \) is the mass of the particle, \( q \) is its charge, and \( B \) is the magnetic field magnitude. Substituting \( v_{\perp} = v \cdot \sin \theta \), we get: \( r = \frac{m \cdot v \cdot \sin \theta}{q \cdot B} \).
03

Determine the Pitch of the Helix

The pitch \( P \) is the distance the particle moves parallel to the magnetic field in one complete revolution. It can be calculated as \( P = v_{\parallel} \cdot T \), where \( T \) is the period of the motion given by \( T = \frac{2 \pi \cdot m}{q \cdot B} \). Thus, \( P = v \cdot \cos \theta \cdot \frac{2 \pi \cdot m}{q \cdot B} \).
04

Find the Ratio of Radius to Pitch

The ratio of the radius to pitch of the helix is calculated as follows:\[\text{Ratio} = \frac{r}{P} = \frac{\frac{m \cdot v \cdot \sin \theta}{q \cdot B}}{v \cdot \cos \theta \cdot \frac{2 \pi \cdot m}{q \cdot B}} = \frac{\sin \theta}{\cos \theta} \cdot \frac{1}{2 \pi}\]Simplifying, the ratio becomes:\[\frac{1}{2\pi} \cdot \tan \theta\]Thus, the correct choice is (d): \( \frac{\tan \theta}{2 \pi} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Charged Particle
A charged particle is an object that carries an electric charge. It could be positive, like a proton, or negative like an electron. When such a particle enters a magnetic field, it experiences a force known as the Lorentz force.
The strength and direction of this force depend on the particle's charge, velocity, and the magnetic field itself. This force causes the particle to move in a path determined by these factors.
  • Positive particles tend to move in one direction.
  • Negative particles move in the opposite direction.
Understanding how charged particles behave in a magnetic field is crucial in physics, as it allows us to predict their movements and apply these principles in various technologies.
Helical Motion
Helical motion refers to a spiral-like path that is formed when a charged particle travels through a magnetic field. This path resembles a spring or coil.
It's a combination of two types of movements: circular and linear. The helical path emerges due to the particular way velocity components act.
  • Circular motion results from the perpendicular component of velocity.
  • Linear motion arises from the parallel component.
These combined elements create the helix, which reflects the inherent dynamics of charged particles within a magnetic field. Such patterns are observed frequently in the fields of electromagnetism and nuclear physics.
Velocity Components
The velocity of a charged particle moving through a magnetic field is split into two distinct components:
  • Perpendicular Component: Denoted as \( v_{\perp} = v \cdot \sin \theta \), this part of the velocity is perpendicular to the magnetic field lines. It drives the particle to move in a circular motion.
  • Parallel Component: Denoted as \( v_{\parallel} = v \cdot \cos \theta \), this part moves the particle along the magnetic field lines, leading to linear motion.
By using trigonometry, these two components help in understanding and predicting the behavior of particles in a magnetic field. They're crucial in modeling their paths and determining parameters like the radius and pitch of the helical trajectory.
Circular and Linear Motion
In the context of magnetic fields, circular and linear motion are essential movements that form the helix path of a charged particle.
Circular motion occurs when a charged particle's perpendicular velocity component makes it rotate around an axis. This is possible because of the constant centripetal force supplied by the magnetic field.
Linear motion, on the other hand, is when the parallel velocity component pushes the particle forward along the magnetic field lines.
  • The combination of these motions results in a helical path.
  • Usually requires a uniform magnetic field for consistent motion.
Recognizing these basic movement patterns allows us to understand more complex scenarios where multiple forces interact.

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Most popular questions from this chapter

Two thin long parallel wires separated by a distance \(b\) are carrying a current \(I\) amp each. The magnitude of the force per unit length exerted by one wire on the other is : (a) \(\frac{\mu_{0} I^{2}}{b^{2}}\) (b) \(\frac{\mu_{0} l^{2}}{2 \pi b}\) (c) \(\frac{\mu_{0} I}{2 \pi b}\) (d) \(\frac{\mu_{0} l}{2 \pi b^{2}}\)

A circular current carrying coil has a radius \(R\). The distance from the centre of the coil on the axis where the magnetic induction will be \(\frac{1}{8}\) th of its value at the centre of the coil, is: (a) \(R \sqrt{3}\) (b) \(\frac{R}{\sqrt{3}}\) (c) \(\frac{2 R}{\sqrt{3}}\) (d) \((2 \sqrt{3}) R\)

Two protons are moving with same velocity in magnetic field of same magnitude, then : (a) magnetic force on protons may be zero (b) magnetic force on both must be same to each other (c) magnetic force on both may or may not be same to each other (d) both (a) and (c) are correct

A positive charge is released from the origin at a place where uniform electric field \(E\) and a uniform magnetic field \(B\) exist along the positive \(y\) -axis and positive \(z\) -axis respectively, then: (a) initially the charge particle tends to move along positive \(z\) -axis (b) initially the charged particle tends to move along negative \(y\) -direction (c) initially the charged particle tends to move along positive \(y\) -direction (d) the charged particle moves in \(y-z\) plane

A coil carrying electric current is placed in uniform magnetic field. Then: (a) torque is formed (b) emf is induced (c) both (a) and (b) are correct (d) none of the above.
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