Electric Particles In Magnetic Field

A diagram shows trajectories of electrons and positrons emitted from a common point $O$ in a uniform magnetic field $\vec{B}$ directed into the page. Five trajectories labeled $a$, $b$, $c$, $d$, $e$ all originate from $O$; trajectories $a$, $b$, $c$ curve to one side ($c$ has the largest radius, $a$ the smallest), while trajectories $d$ and $e$ curve to the opposite side.

A proton moves in a uniform magnetic field of magnitude $B = 0.5$ T along a circular orbit of radius $r = 80$ cm in a plane perpendicular to the field. The proton mass is $m = 1.67 \times 10^{-27}$ kg and its charge is $q = 1.6 \times 10^{-19}$ C.

An electron with kinetic energy $E_k = 10$ eV moves in a circle in a plane perpendicular to a uniform magnetic field of magnitude $B = 10^{-4}$ T. The electron's charge is $e = -1.6 \times 10^{-19}$ C and its mass is $m = 9.1 \times 10^{-31}$ kg.

A device for measuring the charge-to-mass ratio of ions operates as follows. Neutral gas molecules enter an ionization chamber where they are ionized. The ions drift out through slit $S_1$ with negligible initial speed, are accelerated through a potential difference $U$, then enter a uniform magnetic field of magnitude $B$ perpendicular to their velocity through slit $S_3$. The ions undergo uniform circular motion, complete half a revolution, and strike a photographic plate at point $P$ a distance $d$ from $S_3$. The ions carry charge $q$.

When used as a mass spectrometer, ions of the same charge but different masses land at different positions $d$, producing distinct spectral lines. For parts (2) and (3), consider singly-charged magnesium ions (charge $q = 1.6 \times 10^{-19}$ C) accelerated through $U = 2000$ V and entering a magnetic field $B = 50 \times 10^{-3}$ T, with $1\,\mathrm{u} = 1.66 \times 10^{-27}$ kg.

A proton ($P$), a deuteron ($D$), and an alpha particle ($\alpha$), all with the same kinetic energy, enter a uniform magnetic field $\vec{B}$ perpendicular to their velocities. Their masses satisfy $m_\alpha = 2 m_D = 4 m_P$, and their charges are $q_P = q_D = e$, $q_\alpha = 2e$. Let $r_P$, $r_D$, $r_\alpha$ be their respective orbital radii.

A cyclotron has a uniform magnetic field of magnitude $B = 1.5$ T, and the dees have a maximum radius $R = 0.5$ m. Protons are accelerated in this cyclotron. The proton mass is $m = 1.67 \times 10^{-27}$ kg and its charge is $q = 1.6 \times 10^{-19}$ C.

A cyclotron has dees of maximum radius $R = 60$ cm. It is used to accelerate protons from rest to a kinetic energy of $4.0$ MeV. The proton mass is $m = 1.67 \times 10^{-27}$ kg, its charge is $q = 1.6 \times 10^{-19}$ C, and $1$ eV $= 1.6 \times 10^{-19}$ J.

An electron moves along a helical path of radius $R = 20$ cm and pitch $h = 5.0$ cm in a uniform magnetic field of magnitude $B = 2 \times 10^{-3}$ T. The electron's charge-to-mass ratio is $e/m = 1.76 \times 10^{11}$ C/kg.

A magnetohydrodynamic (MHD) generator converts the internal energy of a gas directly into electrical energy. A gas is heated to a very high temperature (above 2500 K) and ionized into a plasma containing both positively and negatively charged particles. The plasma flows with steady speed $v$ between two parallel conducting plates separated by a distance $d$, in a uniform magnetic field $\vec{B}$ that is perpendicular to the flow direction and parallel to the plates. A voltage appears between the two plates.