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Chapter 9: Problem 24

Electrons moving with different speeds enter a uniform magnetic field in a direction perpendicular to the field. They will move along circular paths a. of the same radius h with larger radii for the faster electrons c. with smaller radii for the faster electrons d. either (b) or (c) depending on the magnitude of the magnetic field

Short Answer

Expert verified
b) Larger radii for the faster electrons.

Step by step solution

01

Understanding the Problem

Electrons are entering a uniform magnetic field perpendicularly, which causes them to move in circular paths. We need to determine how the speed of the electrons affects the radius of these circular paths.
02

Formula for Radius of Circular Path

When a charged particle moves in a magnetic field, the force acting on it is the centripetal force causing circular motion. The formula for the radius of the circular path is given by \( r = \frac{mv}{qB} \) where \( m \) is the mass, \( v \) is the velocity, \( q \) is the charge, and \( B \) is the magnetic field strength.
03

Analyzing the Formula

In the formula \( r = \frac{mv}{qB} \), we can see that the radius \( r \) is directly proportional to the velocity \( v \). This means that as the speed (velocity) of the electrons increases, the radius \( r \) will also increase, assuming all other factors like the magnetic field strength \( B \), charge \( q \), and mass \( m \) remain constant.
04

Drawing a Conclusion

From our formula analysis, faster electrons will have larger radii of circular paths. Thus, the correct answer is that the electrons will move along circular paths with larger radii for the faster electrons.

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Key Concepts

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

Electron Motion in Magnetic Field
When electrons enter a magnetic field, they experience a fascinating change in motion. This happens because electrons are charged particles and magnetic fields interact with any moving charges. When electrons, or any charged particles, enter this field perpendicularly, they start moving in circular paths. This kind of motion is another application of the Lorentz force, which is the combination of both electric and magnetic forces on a point charge due to electromagnetic fields.

The reason electrons follow a circular path is because of the perpendicular relationship between their velocity and the magnetic field. In simpler terms, the magnetic force acts as a centripetal force which keeps the electrons moving in a circle. The direction of this force is always changing as the electrons move, always pointing to the center of their circular path.

Understanding this behavior is crucial in technologies such as cyclotrons or mass spectrometers, where precise control of charged particle paths is essential.
Circular Motion of Charged Particles
Charged particles, like electrons, experience a unique form of motion when subjected to a magnetic field, known as circular motion. The magnetic force acts as a centripetal force, meaning it constantly pulls the particle toward the center of its path, allowing the particle to circle around.

This circular path is all due to the right-hand rule, which helps us determine the direction of force acting on a moving charge in a magnetic field. With this rule, we know that the force is perpendicular to both the velocity of the charged particle and the magnetic field, resulting in circular motion. As seen in various applications, this is not just a curios phenomenon but rather a practical principle.

For instance, understanding how charged particles move in circular paths allows us to design cyclotrons for particle acceleration, or control beams of electrons in old cathode-ray tube displays.
Centripetal Force and Magnetic Force
The concepts of centripetal force and magnetic force are closely intertwined when it comes to the motion of charged particles in a magnetic field. When an electron moves through a magnetic field, it experiences a magnetic force that acts perpendicularly to its motion.

This magnetic force effectively becomes the centripetal force, which is required for an object to move in a circle. The key relationship here is expressed in the formula for the radius of the circular path: \[ r = \frac{mv}{qB} \]where:
  • \( r \) is the radius of the path,
  • \( m \) is the mass of the electron,
  • \( v \) is the velocity,
  • \( q \) is the charge of the electron,
  • and \( B \) is the magnetic field strength.
According to this formula, a few things become clear:
  • The radius \( r \) increases with higher velocity \( v \), meaning faster electrons trace larger circular paths.
  • The centripetal force increases either with a stronger magnetic field \( B \) or greater speed \( v \) of the particle.
These insights are crucial for understanding how magnetic confinement works in devices like cyclotrons, where particles must be kept moving at high speeds in tight paths.

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Most popular questions from this chapter

An electron of mass \(m\) is accelerated through a potential difference of \(V\) and then it enters a magnetic field of induction \(B\) normal to the lines. Then, the radius of the circular path is a. \(\sqrt{\frac{2 e V}{m}}\) b \(\sqrt{\frac{2 V m}{e B^{2}}}\) c. \(\sqrt{\frac{2 V m}{e B}}\) d. \(\sqrt{\frac{2 V m}{e^{2} B}}\)

A particle of positive charge \(q\) and mass \(m\) enters with velocity \(V\) ) at the origin in a magnetic field \(B(-\hat{k})\) which is present in the whole space. The charge makes a perfectly inelastic collision with an identical particle (having same charge) at rest but free to move at its maximum positive \(y\) -coordinate. After collision, the combined charge will move on trajectory where \(r=\frac{m V}{q B}\) ) a. \(y=\frac{m V}{q B} x\) b. \((x+r)^{2}+(y-r / 2)^{2}=r^{2} / 4\) c. \((x+r)^{2}+(y-r / 2)^{2}=r^{2} / 8\) d. \((x-r)^{2}+(y+r / 2)^{2}=r^{2} / 4\)

A circular curren carrying coil has a radius \(R\). The distance from the center of the coil on the axis where the magnetic induction will be \((1 / 8)^{\mathrm{h}}\) of its value at the center of the coil, is a. \(R / \sqrt{3}\) b \(R \sqrt{3}\) c \(2 R \sqrt{3}\) d \((2 \sqrt{3}) R\)

A current \(l\) flows a thin wire shaped as regular polygon of \(n\) sides which can be inscribed in a circle of radius \(R\). The magnetic field induction at the center of polygon due to one side of the polygon is a \(\frac{\mu_{0} I}{\pi R}\left(\tan \frac{\pi}{n}\right)\) \& \(\frac{\mu_{0} I}{4 \pi R} \tan \frac{\pi}{n}\) c. \(\frac{\mu_{0} I}{2 \pi R}\left(\tan \frac{\pi}{n}\right)\) d \(\frac{\mu_{0} I}{2 \pi R}\left(\cos \frac{\pi}{n}\right)\)

An infinitely long current carrying wire carries current \(i\). A charge of mass \(m\) and charge \(q\) is projected with speed \(v\) parallel to the direction of current at a distance \(r\) from it, Then, the radius of curvature at the point of projection is a. \(\frac{2 r m v}{q \mu_{0} l}\) b. \(\frac{2 \pi r m v}{q \mu_{0} i}\) c. \(r\) d. cannot be determined
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