D.3 Motion in electromagnetic fields：練習問題

A proton moves in the positive x-direction and enters a region with a uniform magnetic field directed in the positive y-direction. What is the direction of the magnetic force acting on the proton at the moment it enters the field?

An electron enters a uniform magnetic field of strength \(B\) while moving with velocity \(v\). The direction of the velocity makes an angle of \(30^\circ\) with the magnetic field lines. What is the magnitude of the magnetic force acting on the electron?

In a mass spectrometer, ions of isotopes \(X\) and \(Y\) with the same charge \(q\) are accelerated from rest through a potential difference \(V\). They then enter a uniform magnetic field \(B\) perpendicular to their path. The radius of the path for isotope \(X\) is observed to be twice the radius of the path for isotope \(Y\). What is the ratio of their masses \(\frac{m_X}{m_Y}\)?

A particle with charge \( q \) and mass \( m \) enters a uniform magnetic field \( B \) with a velocity \( v \) perpendicular to the field. What is the frequency of the circular motion of the particle?

A proton enters a uniform magnetic field of strength \( B \) at a speed \( v \) perpendicular to the field lines. The proton's path is a circle of radius \( R \). An alpha particle (with charge \( +2e \) and mass \( 4m_p \)) enters the same field at the same speed \( v \) perpendicular to the field lines. What is the radius of the circular path of the alpha particle?

An electron travels horizontally to the right through a region of uniform magnetic field directed vertically upwards. Identify the direction of the magnetic force exerted on the electron at the instant it enters the field.

A proton enters a uniform magnetic field at an acute angle \(\theta\) to the direction of the field lines. Describe the shape of the resulting trajectory and explain the behavior of the velocity component parallel to the field.

In a mass spectrometer, ions of isotopes Magnesium-24 (\(m_1 = 23.99 \text{ u}\)) and Magnesium-26 (\(m_2 = 25.98 \text{ u}\)) are accelerated through a potential difference \(V\) and then enter a region of uniform magnetic field \(B\) perpendicular to their path.

(a) Derive an expression for the separation \(d\) between the two isotopes after they have traveled a semi-circle in the magnetic field.
(b) If \(V = 5.0 \text{ kV}\) and \(B = 0.15 \text{ T}\), calculate the value of \(d\) for singly ionized atoms.
(c) Explain the effect on the separation \(d\) if the potential difference \(V\) is increased.

An electron is accelerated from rest through a potential difference \(V\) and then enters a region of space containing a uniform electric field \(E\) and a uniform magnetic field \(B\) that are mutually perpendicular. The electron's velocity is perpendicular to both fields.

(a) If the electron passes through the fields undeflected, derive an expression for \(V\) in terms of \(E, B, m\), and \(e\).
(b) If the magnetic field is now increased to \(2B\) while \(V\) and \(E\) remain constant, describe the trajectory of the electron and explain your reasoning in terms of forces.
(c) Calculate the radius of the electron's path in the region if only the magnetic field \(B = 5.0 \text{ mT}\) were present, assuming it was accelerated by \(V = 2000 \text{ V}\).