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Ernest Orlando Lawrence and Milton Stanley Livingston reported repeated ion acceleration in 1932. Their evacuated brass chamber used one driven hollow cyclotron dee; the grounded box and slotted divider supplied the other electrode. Filament electrons ionized hydrogen near the gap. A perpendicular magnetic field bent the ions; a high-frequency oscillator drove the gap. Deflecting plates and slits selected particles for a Faraday cup, read by an electrometer.^[ref]

Scanning the magnetic field produced narrow resonance peaks. The optimum deflecting voltage tracked the kinetic energy predicted from field and orbit radius, supporting the reported $1.22\mathrm{MeV}$ protons. Source width and slit geometry limited this check; it was not a direct energy measurement of every particle.^[ref]

This combination distinguishes producing a circulating electric current from establishing its kinetic energy. The electrometer measures collected electric charge per unit time; it cannot by itself tell whether the collected particles are fast. The separate deflection selection supplies a speed-dependent test. Agreement of two consequences of the model, the cyclotron resonance condition and the cyclotron energy and orbit radius relation, makes the inference stronger than either a current signal or a predicted orbit alone.

Milton Stanley Livingston built the early cyclotrons with Ernest Orlando Lawrence; the 1939 Nobel Prize in Physics went to Ernest Orlando Lawrence. His lecture, The evolution of the cyclotron, was delivered on 11 December 1951.^[ref][ref]

An ideal electrical circuit exchanges energy between an inductor and a capacitor. With no electrical resistance or applied drive, the electric charge obeys $L{q}^{\prime \prime }+q/C=0$, giving angular frequency ${\omega }_0=1/\sqrt{LC}$. A series RLC circuit adds dissipation and can include an external drive.

The Lenard cathode-ray window experiment lets cathode rays leave a gas-discharge tube through a thin aluminium foil while preserving the tube's vacuum. This separates production from propagation.

Philipp Lenard's lecture, delivered on 28 May 1906, describes sealing a small hole in a metal plate with aluminium foil. Luminous phosphor grains first rested on the window, then glowed above it in air. His improved tube separated production and observation chambers; coated paper served as a fluorescent screen. An aperture revealed spreading beyond a narrow straight beam. Evacuating the observation chamber reduced spreading and attenuation without interrupting production. Interposed layers and stepped foil stacks gave shadow comparisons; approximate scaling with mass had exceptions.^[ref]

The experimental advantage is unusually clean: reducing the gas content in the observation chamber tests whether gas is needed to carry the cathode rays, while retaining the conditions needed to generate them. The window is a boundary that transmits some electrons while confining the source environment. A spot outside establishes a transmitted effect; the aperture and changed observation environment provide additional ways to distinguish propagation from the source's behavior. This logic is useful whenever a complicated source would otherwise be changed by the very intervention intended to test the emitted beam.

Several inferences require care. A darker fluorescent screen can mean fewer arriving electrons, less deposited energy per electron, or redistribution by electron scattering beyond the observed area. A visible endpoint depends on the screen's sensitivity and background. It is not an intrinsic electron range independent of initial kinetic energy, gas density, and observation geometry. For a modern comparison, measure a reference at fixed source conditions, use background subtraction, and compare the whole luminous patch as well as a small central region. A wider collection area tests redistribution; changing source-to-screen separation also changes collection geometry. These are proposed controls for interpreting a readout, not additional historical observations attributed to Lenard.

Nor is fluorescence a unique signature of cathode rays: X-rays can also excite it. Lenard notes that a later tube could transmit both, and that slow cathode rays failed to pass its window.^[ref] In modern terms, cathode rays are electrons carrying negative electric charge, whereas X-rays are photons with zero electric charge. Magnetic bending of the incident electrons, interpreted through the Lorentz force, supplies a discrimination that brightness alone cannot. Passage through foil alone determines neither the sign of electric charge nor its magnitude, and it does not measure charge-to-mass ratio. The separate Thomson charge-to-mass experiment explains that inference. Even a measured charge-to-mass ratio needs an independent electric charge measurement to determine mass.

A lepton is a Standard Model fermion without color charge. The charged leptons have electric charge $-1$ and their corresponding neutrinos have charge zero. There are three generations; their antiparticles carry opposite electric charge and lepton number. Charged leptons and neutrinos have lepton number $+1$, with the corresponding antiparticles carrying $-1$.

Before calculating electronic energy, ask a simpler question: where can the available valence electrons go? A Lewis structure makes that budget visible, and nitrate shows why a correct budget can still require several resonance structures.

A Lewis structure records which chemical atoms are connected in a molecule or ion and how its valence electrons are allocated. One line represents one bond pair, two lines two bond pairs, and three lines three bond pairs; two dots beside a chemical atom represent one lone pair. The marks are a counting language, rather than measured electron positions. See the IUPAC Gold Book, Lewis formula.

For water, ${\mathrm{H}}_2\mathrm{O}$, the budget is $6+2\times 1=8$ valence electrons. Two O–H lines use four electrons; the other four form two lone pairs on oxygen. For Ammonia, ${\mathrm{NH}}_3$, $5+3\times 1=8$ valence electrons give three N–H lines and one lone pair on nitrogen. Both molecules contain four electron pairs, but their allocations differ. Each shared bond pair is counted once in the total budget, even though both endpoints count it when checking the octet rule.

A practical construction method is:

For carbon dioxide, $4+2\times 6=16$ valence electrons. Starting with two single C–O connections leaves twelve electrons, enough for three lone pairs on each oxygen, but only four electrons around carbon. Converting one lone pair from each oxygen into a second bond pair gives $\mathrm{O}=\mathrm{C}=\mathrm{O}$. Eight electrons are in the four bond pairs, and eight are in four lone pairs, two on each oxygen. Carbon has no lone pair; all three chemical atoms satisfy the octet rule, and all three formal charges vanish.

For the nitrate ion, ${\mathrm{NO}}_3^{-}$, the valence electron budget isNitrogen contributes five, the three oxygen chemical atoms contribute eighteen, and the negative electric charge adds one electron. Start with nitrogen connected to each oxygen by one bond pair: those three lines use six electrons. Put the remaining eighteen electrons into three lone pairs on each oxygen. Each oxygen now satisfies the octet rule, but nitrogen counts only six electrons. Reassign one oxygen lone pair as a second bond pair on its N–O connection. This changes the allocation without adding an electron; it applies the construction procedure in Flowers et al., Chemistry 2e, 7.3: drawing Lewis structures.

The resulting nitrate Lewis structure has one N=O connection and two N–O connections. Its four bond pairs use eight electrons, and its eight lone pairs use sixteen: $8+16=24$. The doubly connected oxygen has two lone pairs; each singly connected oxygen has three; nitrogen has none. All four chemical atoms satisfy the octet rule. Any of the three oxygen chemical atoms can supply the additional bond pair, giving the three equivalent resonance structures below.

Why not draw three N=O connections to make the ion look symmetric? With two lone pairs on each oxygen, that drawing still spends $12+12=24$ electrons and has the right total formal charge, but places twelve electrons around nitrogen. It fails the octet rule for this second-period chemical element. Three equivalent resonance structures express the symmetry without that invalid local count.

The economy is beautiful: a handful of dots and lines rules out many impossible counts and identifies lone pairs that can participate in acid-base chemistry. A Lewis structure helps choose rational candidate connections before a detailed quantum chemistry calculation. It does not specify a complete many-electron wavefunction, actual electron density, or molecular geometry. Adding the separate VSEPR model, described in Chemistry 2e, 7.6: molecular structure and polarity, predicts bent water and pyramidal Ammonia; the layouts above alone make no such prediction. Molecular orbital theory supplies a richer electronic description, including the unpaired electrons of molecular oxygen that an all-paired Lewis structure misses.

In 1916 Gilbert N. Lewis proposed two-dot notation for the shared electrons of a chemical bond and fuller dot formulae for the outer electrons, explicitly relating a shared pair to the existing single-line notation. See Lewis, The Atom and the Molecule, JACS 38, 762–785 (1916), pp. 777–778, available as an ^[ref]; the relevant passage is also readable in a ^[ref]. The modern quantum mechanics interpretation should be distinguished from his historical model of fixed outer electrons.

Linear charge density is electric charge divided by length along a charged line, or locally $dQ/d\ell$. Its SI unit is coulombs per metre. For a uniformly charged long cylindrical conductor, it equals the total surface charge density integrated around the cylinder's circumference; it need not represent a charge literally supported on a one-dimensional line.

The nuclear reactionconserves electric charge and total nucleon count: $3+1=2+2$ and $7+1=4+4$. Its nuclear reaction Q-value is the initial minus final rest energy. It is additional to the incoming beam kinetic energy; the accelerator triggers a rearrangement whose products can carry much more kinetic energy than the incident proton.

Use one consistent set of neutral-atom isotope masses, in unified atomic mass units, from the National Institute of Standards and Technology:These are isotope masses, not element-averaged atomic weights.^[ref] With $\mathrm{u}{c}^2=931.49410372\mathrm{MeV}$,^[ref]The four electron rest masses cancel between sides. Differences of total electronic binding energies do not cancel exactly: this atomic-mass result approximates the bare-nucleus nuclear reaction Q-value at the displayed $0.001\mathrm{MeV}$ scale, not at arbitrary precision. No mixture of rounded nuclear binding energies is needed. At chosen incident kinetic energy $T=0.500\mathrm{MeV}$ on a stationary target, the two outgoing alpha particles together carry approximately $Q+T=17.846\mathrm{MeV}$ before losses in the target and detector window.

The Lorentz force is the force exerted by an electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$ on a body with electric charge $q$ and velocity $\mathbf{v}$. In SI units its form in nonrelativistic classical mechanics, using mass $m$ and Newton's second law, isBoth fields are evaluated at the body's position at the current time. The electric field contributes $q\mathbf{E}$ even at rest, whereas the magnetic force depends on velocity through a cross product. ^[ref]; ^[ref].

For a concrete charged particle in a uniform magnetic field, use nondimensionalization to stipulate $m=q=B=1$, magnetic field $\mathbf{B}=(0,0,1)$, zero electric field, initial position $(0,0,0)$, and initial velocity $(1,0,0)$. These are rescaled quantities, not an assertion that the mass, electric charge, and magnetic field have the same physical units. At the start, $(1,0,0)\times (0,0,1)=(0,-1,0)$, so the magnetic acceleration points toward $-y$. The resulting circular motion isAt $t=\pi /2$ the position is $(1,-1,0)$ and the velocity is $(0,-1,0)$; at $t=\pi$ the position is $(0,-2,0)$ and the velocity is $(-1,0,0)$. Thus the circle has center $(0,-1,0)$ and radius one. Viewed from $+z$ toward the $xy$ plane, the motion is clockwise. Changing only the electric charge to $q=-1$ gives position $(\sin t,1-\cos t,0)$ and reverses the bending to counterclockwise.

Adding initial velocity $v_z=0.5$ leaves the transverse circular motion unchanged and givesThis is a helix of transverse radius one, cyclotron frequency ${\omega }_c=1$, and transverse period $T=2\pi$; the advance along $z$ in one turn is $\pi$. The parallel component of velocity stays constant because the magnetic force has no parallel component: there is no magnetic acceleration along the magnetic field. If the initial velocity were entirely parallel to the magnetic field, the magnetic force would vanish and the path would be straight. ^[ref].

More generally, with zero electric field, no other forces, and a uniform constant magnetic field of magnitude $B>0$, decompose velocity into perpendicular and parallel components. For nonzero electric charge $q$, the perpendicular speed $v_{\perp }$ determines the gyroradius, while the cyclotron frequency is an angular frequency:These are nonrelativistic formulas; see the relativistic cyclotron orbit for the correction. The beauty is that increasing perpendicular speed increases the radius in exactly the proportion needed to leave the period unchanged. Conversely, known magnetic field, perpendicular speed, and measured radius determine the magnitude of the electric charge divided by mass. This makes the same geometry useful for comparing bodies with different electric charge and mass.

Two short checks explain the result. First, the cross product gives $\mathbf{v}\times \mathbf{B}=(-\sin t,-\cos t,0)$ for both stipulated positive-electric charge paths, exactly their acceleration. Second, the transverse magnetic force supplies the centripetal acceleration, so $m{v}_{\perp }^2/r=|q|v_{\perp }B$; for $v_{\perp }>0$ this gives the gyroradius, and ${\omega }_c=v_{\perp }/r$ gives the cyclotron frequency.

Magnetic forces do no work: the instantaneous power of the magnetic part of the Lorentz force vanishes because a cross product is perpendicular to its first argument. The work-energy theorem therefore givesThe dot product $\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})$ is zero at every instant, rather than only after a full turn. With zero electric field and no other forces, the kinetic energy and speed remain constant; the electric field can increase or decrease kinetic energy according to the sign of $q\mathbf{E}\cdot \mathbf{v}$. In the two stipulated examples the constant kinetic energy is respectively $K=1/2$ and $K=5/8$ in rescaled units. The extra parallel velocity raises kinetic energy without changing the gyroradius or the cyclotron frequency.

For an electric example in rescaled units, stipulate $m=q=1$, zero magnetic field, constant electric field $\mathbf{E}=(2,0,0)$, no other forces, and rest at $t=0$. The Lorentz force gives acceleration $(2,0,0)$ and velocity $(2t,0,0)$, hence kinetic energy $K=2t^2$ and power $dK/dt=4t=q\mathbf{E}\cdot \mathbf{v}$. At $t=1$, the kinetic energy has risen from zero to two: the electric field has done positive work.

The existing electromagnetic field tensor formulation expresses the same interaction covariantly. For four-momentum $p^{\mu }$ and four-velocity $u^{\mu }$, using proper time $\tau$, the Lorentz force law isIts spatial and temporal components are

A proteinogenic amino acid represented by K or Lys in protein primary structure. Its amino acid side chain carries predominantly positive electric charge near neutral pH.^[ref]

Trypsin preferentially cleaves after its amino acid residues.

The Madelung constant, named for Erwin Madelung, is a dimensionless geometrical factor in the electrostatic binding of an ionic crystal. For a neutral lattice of electric charges $\pm e$ with nearest unlike-ion distance $r_0$, Coulomb's law gives the potential energy per ion pair as $-{\alpha }_M{e}^2/(4\pi {ϵ}_0{r}_0)$. At a positive reference site, ${\alpha }_M=-{\sum }_{j\ne 0}{z}_j{r}_0/r_j$, with neighboring electric charges $z_{j}e$. Both attraction and repulsion enter this lattice sum; summing with a neutral-bulk prescription is essential. Short-range overlap repulsion and other contributions must be added before inferring a stable spacing or a thermodynamic lattice energy.

For a modern chosen example, the National Institute of Standards and Technology lists neutral magnesium atomic masses $23.985041697\mathrm{u}$, $24.985836976\mathrm{u}$, and $25.982592968\mathrm{u}$, with reference isotopic abundances $0.7899$, $0.1000$, and $0.1101$. They describe the isotopes with mass numbers 24, 25, and 26, respectively.^[ref] These modern inputs are not a reconstruction of historical observations.

Use the dimensionless mass-to-charge ratio coordinate $x=(m/\mathrm{u})/z$. For the illustration only, approximate ion masses by the neutral atomic masses. The singly charged positions are then near $23.98504$, $24.98584$, and $25.98259$; the doubly charged companions are near $11.99252$, $12.49292$, and $12.99130$. The neutral-to-ion correction already given above is essential at higher accuracy. A peak near $x=12$ alone need not identify a chemical element: another ion with a different electric charge can occupy the same mass-to-charge ratio interval.

For the narrow singly charged peak at $x\approx 24.98584$, the chosen full width at half maximum $\Delta x=0.12$ gives mass resolving power $R\approx 208.2$; widening to $\Delta x=1.20$ gives $R\approx 20.82$. Adjacent isotope spacings here are approximately one unit in $x$. Peaks are visibly separated in the first chosen profile and overlap strongly in the second. A small uncertainty in a fitted peak centre would not by itself make the overlapping components resolved.

Even integrated peak areas need a response model. For isotope fraction $f_i$, let ${\eta }_{i,z}$ collect the probability of ionization into electric charge state $z$, transmission through the analyzer, and the detection response. In a linear instrument with overall exposure factor $C$, the component area isEqual ${\eta }_{i,z}$ at a fixed $z$ makes normalized areas equal the fractions; otherwise it does not. In the chosen example, doubling only the mass-25 response changes the normalized areas to $0.71809$, $0.18182$, and $0.10009$, although the neutral sample is unchanged. Photographic darkening introduces another possible nonlinearity: neither line darkness nor peak height is a universal measure of neutral isotopic abundance. Calibration of position and calibration of intensity answer different questions. Matched standards, a known electric charge assignment, and a response correction make isotope analysis useful; a visually impressive set of peaks alone does not supply those controls.

A conducting conical pendulum connected to a grounded pivot develops a motional electric potential difference $B\omega r^2/2$ between the rotation axis and a bob at radius $r$. If the bob dominates the capacitance, its electric charge is $CB\omega r^2/2$. An additional uniform vertical electric field changes both the force on that charge and, while the grounding connection remains, the background electric potential entering the charging relation. These effects must be treated consistently when calculating a first-order mechanical response.

A magnetic field is a vector field $\mathbf{B}(\mathbf{r},t)$ whose local value determines the magnetic part of the Lorentz force on a body with electric charge $q$ and velocity $\mathbf{v}$. In SI units,where $v$ is the speed, $B=|\mathbf{B}|$, and $\alpha$ is the angle between the velocity and the magnetic field. The cross product makes a nonzero magnetic force perpendicular to both the velocity and the magnetic field. The right-hand rule gives its direction for positive electric charge; negative electric charge reverses it. ^[ref].

The SI unit of magnetic field is the tesla:For a stipulated example in SI units, take electric charge $q=2\mathrm{C}$, velocity $\mathbf{v}=(3,0,0)\mathrm{m}{\mathrm{s}}^{-1}$, and magnetic field $\mathbf{B}=(0,0,4)\mathrm{T}$. The cross product gives magnetic force ${\mathbf{F}}_B=(0,-24,0)\mathrm{N}$. With the same speed directed along $+z$, the magnetic force is zero: motion parallel to a magnetic field produces no magnetic acceleration.

Magnetic field lines visualize the direction of a magnetic field by being tangent to it at each point. They are not generally the path of a body with electric charge: in a uniform magnetic field along $+z$, the magnetic field lines are straight while transverse velocity produces circular motion, and adding parallel velocity produces helical motion in a uniform magnetic field. The worked Lorentz force example below illustrates this distinction.

A magnetic field need not be static. Magnetostatics is the restricted case with no dependence on time; an electromagnetic wave has changing electric field and magnetic field even in vacuum. A changing magnetic field is coupled to the electric field through Faraday's law of induction. ^[ref]. The useful separation is between local bending, described by the Lorentz force, and the amount of magnetic field crossing a surface, described by magnetic flux.

Magnetic rigidity measures the resistance of a charged particle's momentum to bending by a magnetic field. The relativistic cyclotron orbit gives $p_{\perp }=|q|Br$, where $r$ is the transverse gyroradius. For motion perpendicular to a uniform magnetic field, $p_{\perp }=p$ is the whole momentum. A measured magnetic rigidity fixes momentum per magnitude of electric charge; it does not by itself fix mass, kinetic energy, or the sign of the electric charge. This distinction makes the Anderson positron track experiment an especially revealing application of the Lorentz force.

A spectrometer that selects charged particles using the bending of their trajectories in a magnetic field. For a uniform magnetic field of magnitude $B$, the perpendicular momentum component is $p_{\perp }=|q|Br$, where $q$ is the electric charge and $r$ is the curvature radius of the projected trajectory.

Let $m_z={\int }_V{M}_{z}dV$ be the axial magnetic dipole moment, where $M_z$ is the axial magnetization. Write $J_z$ for the internal angular momentum associated with that magnetic dipole moment, and $I\dot{\theta }$ for the body's rotational angular momentum, with moment of inertia $I$. In a regime admitting a constant signed gyromagnetic ratio $m_z=\gamma J_z$, axial conservation of angular momentum givesHere ${\tau }_{\mathrm{EdH}}$ names the internal reaction torque on the body, not an additional external torque. The external term includes the suspension. The model assumes that the changing internal angular momentum is transferred to the measured mechanical motion, with other storage or escape channels negligible on the observation timescale. A changing or non-collinear mixture of orbital angular momentum and spin angular momentum need not admit a single constant $\gamma$.

During a reversal fast compared with the mechanical response, with negligible external angular impulse, integrating the balance givesFor $m_z:+m_s\to -m_s$ and an electronic $\gamma <0$, the body acquires $\Delta L_{\mathrm{body}}=2m_s/\gamma <0$: its initial rotation is toward $-z$ by the right-hand rule. Reversing the reversal reverses the impulse. Over a complete periodic cycle, $\Delta m_z=0$, so the net reaction angular impulse vanishes; a resonant alternating response does not imply steadily accumulating angular momentum.

The orbital angular momentum prediction can be derived without a material model. A charge $q$ traveling around a circle of radius $r$ at angular frequency $\omega$ has loop electric current $q\omega /(2\pi )$ and magnetic dipole moment $m=q\omega r^2/2$. Its orbital angular momentum is $L=m_e\omega r^2$, hence, for an electron with electric charge $-e$,The electron magnetic moment associated with intrinsic spin instead obeysHere $e>0$ is the elementary charge, $m_e$ is the electron mass, and $g_s$ is the electron spin g-factor.^[ref] Thus the same change of magnetic dipole moment corresponds to about half as much transferred angular momentum in the pure-spin model as in the orbital angular momentum model. This factor of two is a relation between magnetic dipole moment and angular momentum, not evidence that an electron is literally a rotating charged sphere. In a material with both contributions, the inferred coefficient is a material-dependent magnetomechanical gyromagnetic ratio.

A physicist and chemist whose investigation of radioactivity joined sensitive measurement to chemical separation. She moved from Warsaw to Paris to study, later becoming a professor of physics. The Curie ionization-current measurement shows the experimental craft behind the work: balancing a tiny electric current against a calibrated electric charge. She shared the 1903 Nobel Prize in Physics and received the 1911 prize in chemistry.^[ref]

Mass spectrometry separates and detects ions according to mass-to-charge ratio. It can distinguish isotopes which belong to the same chemical element but have different atomic masses. The instrument measures ions: interpreting the result as a neutral atomic mass requires accounting for electron removal and its ionization energy. For a singly positive ion in its ground state, $m_{+}=m_{\mathrm{a}}-m_e+I/c^2$, where $m_e$ is the electron rest mass, $I$ is the ionization energy, and $c$ is the speed of light.

For a simple illustrative analyzer, positive ions of mass $m$ and electric charge $q=ze$ gain kinetic energy $qV=m{v}^2/2$ across an electric potential difference $V$ before entering a transverse uniform magnetic field $B$. The Lorentz force supplies centripetal acceleration, soAt fixed $V$ and $B$, the measured bending radius $r$ separates mass-to-charge ratios, not masses independently of electric charge. This ideal model assumes negligible initial kinetic energy and speeds small compared with the speed of light; the historical focusing mass spectrograph used the geometry described below.

The experimental achievement is to turn invisible differences between individual ions into separated, calibrated positions. Aston velocity focusing explains how an early mass spectrograph sharpened its images; the velocity-selected magnetic mass analyzer provides an independent model whose equations can be checked directly. The magnesium isotope mass spectrum shows why position, width, and intensity carry different information.

A mass spectrum displays detected signal against mass-to-charge ratio. Peak positions identify candidate ions; peak areas give isotopic abundances only after the response and overlapping contributions have been accounted for.

Broadening can hide a weak component without changing its area in this ideal model. An actual mass spectrum can also contain molecular ions, different electric charge states, background, and unequal detection responses. A missing visible peak therefore cannot by itself prove the absence of an isotope. Conversely, a fractional atomic weight is a property of a population: there is no requirement for an ion peak at the weighted mean.

The ratio of an ion's mass to its electric charge; in SI it has units $kg/C$. A convenient numerical mass spectrum coordinate is $x=(m/\mathrm{u})/z$, with $q=ze$ and positive integer $z$ here. This $x$ is dimensionless. Doubling $z$ at fixed mass halves $x$, so the electric charge state must be identified before assigning an isotope.