MOTION IN A MAGNETIC FIELD, as detailed in Chapter Four of the NCERT textbook "MOVING CHARGES AND MAGNETISM" for Class 12 Physics (New Edition), is crucial for students preparing for competitive exams like JEE, and NEET. This topic is covered comprehensively in NCERT and RBSE curricula.
MOTION IN A MAGNETIC FIELD
In a magnetic field, the force on a moving charge is perpendicular to its velocity. As a result, there is no work to be done and no change in speed. This differs from an electric field force, which can transfer energy and change the velocity of charge by acting parallel or antiparallel to its motion.
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The magnetic force on a moving charge in a magnetic field is always perpendicular to its velocity. Therefore, it does not work on the particle and does not change the magnitude of the particle's velocity.
This is unlike the force due to an electric field, qE which can have components parallel to motion and thus can transfer energy in addition to momentum.
Motion in a Uniform Magnetic Field
Circular Motion
When the velocity of a charged particle is perpendicular to the magnetic field B the magnetic force acts as a centripetal force. As a result, the particle moves in a circular path.
Circular Motion In Magnetic Field
Helical Motion
If the particle's velocity has a component parallel to the magnetic field, this component remains unchanged, and the particle follows a helical path. The motion is a combination os circular motion in the plane perpendicular to `\vec B` and linear motion along `\vec v`.
Helical Motion
Centripetal Force and Motion in a Magnetic Field
☆ Centripetal force: The force acting on a particle moving in a circular path with radius r is given by
`F = \frac {mv^2}{r}`
This force is directed towards the center of the circle and is called the centripetal force.
☆ Magnetic force: When the velocity v of the particle is perpendicular to a magnetic field B, the magnetic force also acts as a centripetal force. The magnitude of the magnetic force is given by
`F = q v B`
Equating the centripetal force to the magnetic force
`q v B = \frac {m v^2}{r}`
`r = \frac {m v^2}{q v B}`
`r = \frac {m v}{q B}`
☆ Conclusion: The radius r is proportional to the particle's momentum (p = m v). Thus, the larger the momentum, the larger the radius of te circular path.
☆ If `\omega` is the angular frequency, then `v = \omega r`.
Therefore
`r = \frac {m \omega r}{q B}`
`\omega = \frac {q B}{m}` ........(1)
`2 \pi \nu = \frac {q B}{m}`
` \nu = \frac {q B}{2 \pi m}` ........(2)
Where `\nu` is the frequency of rotation.
☆ The angular frequency `omega` and rotational frequency `\nu` are independent of the particle's velocity or energy. This principle is use in design of a cyclotron, where the frequency `nu` is constant regardless of the particle's energy.
Time for One Revolution
From eq. (1)
`\omega = \frac {q B}{m}`
`\frac {2 \pi}{T} = \frac {q B}{m}`
`\frac {T}{2 \pi} = \frac {m}{q B}`
` T = \frac {2 \pi m}{q B}`
Component of velocity Parallel to Magnetic Field
If there is a component of the velocity parallel to the magnetic field (denoted by `v_{11}`). It will make particles move along the field and the particles would be helical.
Pitch (p) of the Helical Path
The distance moved along the magnetic field in one rotation is known as one rotation.
Formula of Pitch
`p = \text{velocity} \times \text {Time for one revolution}`
`p = v_{11} \times T`
`p = \frac{2 \pi m}{q B} v_{11} `
Radius of the Helix
The radius of the circular component of motion is called the radius of the helix.
Questions and Answers
1. What is the direction of the force on a moving charge in a magnetic field relative to its velocity?
Ans. The force is perpendicular to the velocity of the moving charge.
2. Why does a magnetic force not change the speed of a moving charge?
Ans. Because the force is perpendicular to the velocity, no word is done and no change in speed occurs.
3. What type of motion does a charged particle exhibit when its velocity is perpendicular to a uniform magnetic field?
Ans. The particle moves in a circular path.
4. What is the formula for the centripetal force acting on a particle moving in a circular path of radius r?
Ans. Formula of Centripetal Force
`F = \frac {m v^2}{r}`
5. What is the formula of magnetic force on a particle moving perpendicular to a magnetic field?
Ans. Formula of Magnetic Force
`F = q v B`
6. Write the formula for the charged particle's angular frequency `\omega` in a magnetic field.
Ans. The formula for the charged particle's angular frequency is
`\omega = \frac {q B}{m}`
7. How is the frequency of rotation `\nu` related to the angular frequency `omega`?
Ans. The relation between frequency of rotation and angular frequency is
` \nu = \frac {q B}{2 \pi m}`
8. What is the time period T for one revolution of a charged particle in a magnetic field.
Ans. The formula of time period
` T = \frac {2 \pi m}{q B}`
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