The
charge-to-mass ratio (
Q/
m) of an object is, as its name implies, the
charge of an object divided by the mass of the same object. This quantity is generally useful only for objects that may be treated as particles. For extended objects, total charge, charge density, total mass, and mass density are often more useful. Derivation: qvB = mv\frac{v}{r} or {{NumBlk||\frac{q}{m}=\frac{v}{Br}|}} Since F_\text{electric} = F_\text{magnetic}, E q = B q v or {{NumBlk||v = \frac{E}{B}|}} Equations () and () yield \frac{q}{m}=\frac{E}{B^2r}
Significance In some experiments, the charge-to-mass ratio is the only quantity that can be measured directly. Often, the charge can be inferred from theoretical considerations, so the charge-to-mass ratio provides a way to calculate the mass of a particle. Often, the charge-to-mass ratio can be determined by observing the deflection of a charged particle in an external
magnetic field. The
cyclotron equation, combined with other information such as the
kinetic energy of the particle, will give the charge-to-mass ratio. One application of this principle is the mass spectrometer. The same principle can be used to extract information in experiments involving the
cloud chamber. The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. It turns out that gravitational forces are negligible on the subatomic level, due to the extremely small masses of subatomic particles.
Electron The electron charge-to-mass quotient, -e/m_{e}, is a quantity that may be measured in experimental physics. It bears significance because the electron mass
me is difficult to measure directly, and is instead derived from measurements of the elementary charge
e and e/m_{e}. It also has historical significance; the
Q/
m ratio of the electron was successfully calculated by
J. J. Thomson in 1897—and more successfully by Dunnington, which involves the
angular momentum and deflection due to a perpendicular
magnetic field. Thomson's measurement convinced him that
cathode rays were particles, which were later identified as
electrons, and he is generally credited with their discovery. The
CODATA recommended value is CODATA refers to this as the
electron charge-to-mass quotient, but
ratio is still commonly used. There are two other common ways of measuring the charge-to-mass ratio of an electron, apart from Thomson and Dunnington's methods. • The magnetron method: Using a GRD7 Valve (Ferranti valve), electrons are expelled from a hot tungsten-wire filament towards an anode. The electron is then deflected using a solenoid. From the current in the solenoid and the current in the Ferranti Valve, e/m can be calculated. • Fine beam tube method: A heater heats a cathode, which emits electrons. The electrons are accelerated through a known potential, so the velocity of the electrons is known. The beam path can be seen when the electrons are accelerated through a helium (He) gas. The collisions between the electrons and the helium gas produce a visible trail. A pair of
Helmholtz coils produces a uniform and measurable magnetic field at right angles to the electron beam. This magnetic field deflects the electron beam in a circular path. By measuring the accelerating potential (volts), the current (amps) to the Helmholtz coils, and the radius of the electron beam, e/m can be calculated.
Zeeman Effect The charge-to-mass ratio of an electron may also be measured with the
Zeeman effect, which gives rise to energy splittings in the presence of a
magnetic field B: \Delta E = \frac{e\hbar B}{2m}(m_{j,f}g_{J,f}-m_{j,i}g_{J,i}) Here
mj are quantum integer values ranging from −
j to
j, with
j as the
eigenvalue of the
total angular momentum operator J, with : \mathbf{J} = \mathbf{L} + \mathbf{S} where
S is the
spin operator with eigenvalue
s and
L is the
angular momentum operator with eigenvalue
l.
gJ is the
Landé g-factor, calculated as g_J = 1 + \frac{j(j+1) + s(s+1) - l(l+1)}{2j(j+1)} The shift in energy is also given in terms of
frequency υ and
wavelength λ as \Delta E = h\Delta\nu = h c \Delta \left( \frac{1}{\lambda} \right ) = hc \frac{\Delta\lambda}{\lambda^2} Measurements of the Zeeman effect commonly involve the use of a
Fabry–Pérot interferometer, with light from a source (placed in a magnetic field) being passed between two mirrors of the interferometer. If
δD is the change in mirror separation required to bring the
mth-order ring of wavelength into coincidence with that of wavelength
λ, and Δ
D brings the ring of wavelength
λ into coincidence with the
mth-order ring, then \Delta\lambda = \lambda^2\frac{\delta D}{2D\Delta D}. It follows then that hc\frac{\Delta\lambda}{\lambda^2} = hc\frac{\delta D}{2D\Delta D} = \frac{e\hbar B}{2m}(m_{j,f}g_{J,f}-m_{j,i}g_{J,i}) \, . Rearranging, it is possible to solve for the charge-to-mass ratio of an electron as \frac{e}{m} = \frac{4 \pi c}{B(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})}\frac{\delta D}{D\Delta D} \, . == See also ==