Motion of Charged Particle in Magnetic Field - UNSOLVED PRACTICE SET

Q7. Derive the expression for the radius of the circular path of a charged particle moving perpendicular to a uniform magnetic field.

Q8. Show that the time period of revolution of a charged particle in a uniform magnetic field is independent of its velocity and radius.

Q9. A proton moves in a circular path of radius 10 cm in a uniform magnetic field of 0.5 T. Calculate its speed and kinetic energy.

[Given: mₚ = 1.67 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C]

Q10. Why does the kinetic energy of a charged particle remain constant when it moves in a uniform magnetic field? Explain.

Q11. An electron moves in a magnetic field with velocity having components v_parallel = 2 × 10⁶ m/s and v_perpendicular = 3 × 10⁶ m/s. The magnetic field B = 0.2 T. Describe the path and calculate the pitch of the helix.

[Given: mₑ = 9.1 × 10⁻³¹ kg, e = 1.6 × 10⁻¹⁹ C]

Q12. What is a magnetic mirror? How does it help in confining charged particles?

Q13. Explain the motion of a charged particle in a uniform magnetic field when it enters (a) perpendicular to the field, and (b) at an angle θ to the field. Draw diagrams showing the paths in both cases and derive the relevant expressions.

Q14. Explain the principle of a magnetic bottle (magnetic mirror). How is this principle used to confine plasma in fusion reactors? Why is this confinement necessary?

Q15. An electron is accelerated through a potential difference of 1000 V and then enters a uniform magnetic field B = 0.2 T.

(a) Calculate the velocity of the electron.  

(b) If it enters perpendicular to the field, calculate the radius of its path.  

(c) Calculate the time period and frequency of revolution.  

(d) If it enters at 60° to the field, calculate the radius of the helical path and the pitch.

[Given: mₑ = 9.1 × 10⁻³¹ kg, e = 1.6 × 10⁻¹⁹ C]

Q16. In a school demonstration, a teacher shows the path of electrons in a magnetic field using a vacuum tube (Teltron tube). Electrons are accelerated by a potential difference of 250 V and then enter a uniform magnetic field perpendicular to their velocity.

(a) Calculate the speed of the electrons.  

(b) The teacher adjusts the magnetic field so that the electrons trace a circle of diameter 8 cm. Calculate the magnetic field strength.  

(c) If the accelerating voltage is increased to 500 V, what magnetic field is needed to maintain the same path?  

(d) Calculate the time taken by the electrons to complete one circle in both cases. What do you observe?  

(e) Explain why this time period independence is called "isochronism" and why it's important in cyclotrons.

[Given: mₑ = 9.1 × 10⁻³¹ kg, e = 1.6 × 10⁻¹⁹ C]

Q17. In the Van Allen radiation belts around Earth, charged particles are trapped by Earth's magnetic field and move in helical paths.

(a) Explain why the particles move in helical paths rather than circles or straight lines.  

(b) A proton in the Van Allen belt has v_parallel = 1 × 10⁶ m/s and v_perpendicular = 2 × 10⁶ m/s. Earth's magnetic field at that location is B = 3 × 10⁻⁵ T. Calculate the radius and pitch of the helix.  

(c) The particles are reflected at points where the magnetic field is stronger (magnetic mirrors). Explain this phenomenon.  

(d) Calculate the time between successive reflections if the distance between mirror points is 1000 km.  

(e) Why are the Van Allen belts important for protecting life on Earth?

[Given: mₚ = 1.67 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C]

Q18. In a mass spectrometer, ions of different masses are separated by their motion in a magnetic field. Singly ionized atoms of an element (charge +e) are accelerated through 5000 V and enter a uniform magnetic field B = 0.8 T perpendicular to their velocity.

(a) Show that the radius of the path is proportional to the square root of the mass.  

(b) Two isotopes have masses 35 u and 37 u (1 u = 1.66 × 10⁻²⁷ kg). Calculate the separation between their impact points on a photographic plate placed 180° around the magnet.  

(c) If the magnetic field has a slight non-uniformity of 1%, what is the percentage error in mass determination?  

(d) Explain why mass spectrometers need high vacuum and why the ions must be singly charged for simplest analysis.  

(e) A student suggests using an electric field instead of a magnetic field for separation. Compare the two methods.