Angular magnification is the ratio of the signed viewing angle with an optical instrument to the signed viewing angle in a stated reference arrangement. Measure both as apparent directions from the eye, positive toward the same side of the optical axis; then an inverted view has negative $M_{\theta }$. If only positive angular sizes are compared, the result is $|M_{\theta }|$ and carries no orientation information. ^[ref].

For the upright virtual image of a converging thin lens, let the eye be at $x=e\ge 0$ and take the unaided reference to be the same small object at a distance $D$ from the eye. Within the small-angle approximation,Specify $D$: keeping the object in its original position means $D=e+u$, whereas the customary magnifier rating uses a near-point reference, often $D=25\mathrm{cm}$. A reference distance is a convention, not a claim that every eye has that near point. ^[ref].

For the virtual image diagram, $m=4$, $v=-30\mathrm{cm}$ and $e=5\mathrm{cm}$. With $D=25\mathrm{cm}$, $M_{\theta }\simeq 20/7$, whereas keeping the object at $D=e+u=12.5\mathrm{cm}$ gives $M_{\theta }\simeq 10/7$. Thus neither viewing comparison gives the transverse magnification four. A person may also be unable to focus on the unaided object at the shorter reference distance.

At $u=f$, each outgoing bundle is parallel with slope $-h_o/f$; its apparent viewing direction has angle ${\theta }_i\simeq h_o/f$. Consequently $M_{\theta }\simeq D/f$ remains finite even though $m$ is undefined at a finite image plane. Equivalently, in the limit $u\to f^{-}$, the factors $m$ and $e-v$ both grow without bound but their ratio tends to $1/f$. For $f=10\mathrm{cm}$ and $D=25\mathrm{cm}$, the relaxed-view prediction is $M_{\theta }\simeq 2.5$. ^[ref].

Transverse magnification is the ratio of corresponding signed lengths perpendicular to the optical axis. For an ideal image of a small arrow of nonzero height,Use the same upward-positive height convention in both planes. Thus an upright $0.50\mathrm{cm}$ arrow imaged as $-0.25\mathrm{cm}$ has $m=-1/2$: inverted and half as tall. The sign describes orientation; real image versus virtual image describes ray intersections. They are separate properties, even when a particular thin lens setup correlates them.^[ref]

For a thin lens with object coordinate $-u$ and image coordinate $v$, the undeviated centre light ray is the straight line through $(-u,h_o)$ and $(0,0)$, so its height at $x=v$ is $h_i=-h_{o}v/u$. This is the signed similar triangles argument. Combining it with the thin lens equation givesFor $f>0$, $u>2f$ gives $-1<m<0$; $u=2f$ gives $m=-1$; $f<u<2f$ gives $m<-1$; and $0<u<f$ gives $m>1$. At $u=f$, the image is conventionally “at infinity”: the divergent expressions for $v$ and $m$ describe the limiting behaviour, not a finite image of infinite height. Different object points produce parallel bundles in different directions, which is why angular magnification remains meaningful.

The real image experiment tests $m$ in two independent ways: signed ratios of measured corresponding intervals, and $-v/u$ from measured longitudinal distances. As a further prediction, moving from $u=30\mathrm{cm}$ to $u=15\mathrm{cm}$ for $f=10\mathrm{cm}$ swaps $u,v$, changing $m=-1/2$ to $m=-2$. The two image sizes differ by a factor of four, while both images remain inverted. This reversal follows from the symmetric thin lens equation and connects to the Bessel lens displacement method.

Under proper rotations a third-rank isotropic tensor is proportional to the Levi-Civita symbol. Its contraction with a field vector is antisymmetric in the two stress indices. A physical symmetric stress therefore vanishes in this linear constitutive model. Under the full orthogonal group an ordinary isotropic third-rank tensor is already zero; the orientation-sensitive Levi-Civita tensor requires the usual pseudotensor distinction.

For an axisymmetric magnetic field $\mathbf{B}=\psi {\mathbf{e}}_{\varphi }/(r\sin \theta )+\nabla \times [\chi {\mathbf{e}}_{\varphi }/(r\sin \theta )]$, the poloidal magnetic flux function gives $B_r={\chi }_{\theta }/(r^2\sin \theta )$ and $B_{\theta }=-{\chi }_r/(r\sin \theta )$. Force balance implies locally $p=p(\chi )$ and $\psi =\psi (\chi )$ on regular connected flux surfaces. Direct calculation gives $(\nabla \times \mathbf{B})\times \mathbf{B}=-({\Delta }_{\ast }\chi +\psi {\psi }^{\prime })\nabla \chi /(r^2{\sin }^2\theta )$, with ${\Delta }_{\ast }\chi ={\chi }_{rr}+\sin \theta {\partial }_{\theta }({\chi }_{\theta }/\sin \theta )/r^2$. Equating this to ${\mu }_0\nabla p$ proves the displayed magnetostatic equilibrium equation, including its sign and dimensions.

An induced magnetosphere is maintained by the interaction of an external magnetized plasma flow with a conducting ionosphere or atmosphere. Its currents and draped magnetic field do not establish an internal planetary dynamo. Venus is a central example; Mars combines this interaction with localized crustal fields.

The Lehnert number compares an Alfvén speed to a rotation speed on length scale $L$. With $\mathrm{Le}\ll 1$, magnetic-wave frequencies are small compared with the rotation frequency, allowing nearly geostrophic flow. Ideal weakly damped magnetic waves also require $\eta /L^2$ and $\nu /L^2$ to be small compared with their wave frequency; a small Lehnert number alone does not guarantee weak diffusion.

An incompressible Boussinesq fluid with uniform magnetic field, uniform vertical rotation and uniform temperature gradient supports waves combining magnetic tension, Coriolis force and buoyancy. Put $K=|\mathbf{k}|$, $h^2=K^2-k_z^2$, $a^2=(\mathbf{B}\cdot \mathbf{k}{)}^2/({\mu }_0\rho )$ and $H=g\alpha \beta$, with ${\partial }_z{T}_0=-\beta$. Eliminating pressure and the temperature and magnetic perturbations gives the displayed dispersion relation for nonzero frequencies. Stable thermal stratification has $H<0$. Clearing frequency denominators requires checking any resulting zero roots in the original equations; a stationary magnetostatic-buoyancy balance can exist independently of the propagating branches.

Electron magnetohydrodynamics neglects ion motion and electron inertia while retaining transport of the magnetic field by the electron current. At constant electron number density, ${\mathbf{u}}_e=-\mathbf{J}/(en)=-c\nabla \times \mathbf{B}/(4\pi en)$. Flux freezing in this velocity givesFor a uniform guide field, the solenoidal linear equation implies ${\omega }^2=(c{B}_0/4\pi en{)}^2{k}_{\parallel }^2{k}^2$. With $v_A=B_0/\sqrt{4\pi n{m}_i}$ and $d_i=c\sqrt{m_i/(4\pi e^{2}n)}$, the coefficient is $c{B}_0/(4\pi en)=v_A{d}_i$.

Hall magnetohydrodynamics retains bulk ion inertia but freezes magnetic flux into the electron velocity ${\mathbf{u}}_e=\mathbf{u}-\mathbf{j}/(en)$. With constant density, negligible electrical resistivity and electron inertia, Gaussian units give $\mathbf{j}=c\nabla \times \mathbf{B}/(4\pi )$ and ${\partial }_t\mathbf{B}=\nabla \times [(\mathbf{u}-c\nabla \times \mathbf{B}/(4\pi en))\times \mathbf{B}]$. The ion velocity still satisfies the magnetohydrodynamic momentum equation. Neglecting that velocity gives electron magnetohydrodynamics, a further approximation rather than a synonym for the complete Hall system.

A theta pinch is a cylindrical plasma configuration with axial magnetic field and azimuthal electric current. For a static equilibrium without gravity, the radial Lorentz force gives $d[p+B^2/(8\pi )]/dr=0$ in Gaussian units. This straight-field configuration has no field-line curvature drive of the usual interchange instability.

A kinematic magnetic dynamo studies magnetic field amplification by a prescribed fluid flow, neglecting the field's feedback on the velocity through the Lorentz force. With incompressibility and zero magnetic diffusivity, a Lagrangian field vector evolves by its local velocity-gradient matrix. A growth law derived in this approximation cannot hold indefinitely once magnetic stresses significantly change the flow.

Mean-field electrodynamics applies averaging and a response closure to magnetic induction in a fluctuating conducting fluid. The mean equation contains the correlated mean-field electromotive force in the title. Scale separation permits a local expansion into alpha transport, turbulent diffusion and higher gradients; general anisotropic or finite-memory statistics require tensorial and nonlocal responses. A mean-field dynamo is a self-exciting application of this framework, not a consequence of averaging alone.

The leading force balance in a rapidly rotating conducting fluid when inertial and viscous forces are small, while Coriolis force, pressure, buoyancy and Lorentz force density remain significant. The Rossby number and Ekman number measure the omitted inertia and bulk viscosity. Rotation does not balance an arbitrary magnetic torque: impenetrable axisymmetric boundaries impose the Taylor constraint. Magnetic diffusion can remain in the induction equation even when viscosity is neglected in momentum balance.

Magnetic flux concentration redistributes existing magnetic flux into a smaller area. For a locally straight flux bundle with conserved flux, $B\mathcal{A}$ is constant, so decreasing area increases field strength. Converging convection, field-line stretching and flux expulsion can produce concentrations; finite magnetic diffusivity opposes them and Lorentz force feedback limits the kinematic approximation. Redistribution alone is not evidence of net dynamo action.

Alfvénic turbulence is the turbulent interaction of fluctuations polarized primarily as Alfvén waves. In Elsässer variables ${\mathbf{z}}^{\pm }={\mathbf{u}}_{\perp }\pm \delta {\mathbf{B}}_{\perp }/\sqrt{4\pi {\rho }_0}$, each wave population is nonlinearly distorted by the oppositely propagating population. Equal populations give a balanced cascade; unequal populations give an imbalanced cascade. The absence of one population removes the leading nonlinear interaction in ideal reduced magnetohydrodynamics.

Critical balance is a scale-dependent ordering in which the wave-propagation time and nonlinear interaction time are comparable. For strong Alfvénic turbulence, $k_{\parallel }{v}_A\sim k_{\perp }\delta u_{\perp }$. For electron magnetohydrodynamics, the linear rate is $v_A{d}_i{k}_{\parallel }{k}_{\perp }$ and the nonlinear rate is $v_A{d}_i{k}_{\perp }^2\delta B/B_0$, so critical balance instead gives $\delta B/B_0\sim k_{\parallel }/k_{\perp }$. This assumption is an estimate of decorrelation dynamics, not an exact equality at every point.

An electron-magnetohydrodynamic cascade transfers magnetic fluctuations through the nonlinear electron magnetohydrodynamics induction equation. Writing $b_l=\delta B_l/B_0$, its perpendicular nonlinear interaction time is ${\tau }_{\mathrm{nl}}\sim l_{\perp }^2/(v_A{d}_i{b}_l)$. The anisotropic whistler wave time is ${\tau }_w\sim l_{\parallel }{l}_{\perp }/(v_A{d}_i)$. Scaling predictions depend on locality, phase decorrelation and the assumed balance of these times.

In a local isotropic hydrodynamic inertial-range model with only $ϵ$ and $k$, dimensional analysis fixes the Kolmogorov 1941 theory spectrum. A mean magnetic field supplies the independent Alfvén speed, permitting the arbitrary dimensionless function in the displayed formula even before anisotropy adds parallel scales. Cascade-time assumptions such as weak decorrelated interactions or critical balance are therefore necessary to select an MHD spectrum.

Remanent magnetization is magnetization retained after the applied magnetic field has been removed. Magnetic minerals cooling in an ambient field can preserve information about that field in a planetary crust. Such a record may outlive the planetary dynamo which originally produced it. Crustal remanence therefore differs from a present field maintained by conducting-fluid motion.

The Curie temperature is the transition temperature above which a ferromagnetic material loses spontaneous magnetization. A hot conducting planetary core can generate a magnetic field through dynamo action without behaving as a permanent magnet. The blocking temperature at which a mineral records crustal remanence also depends on its relaxation timescale and need not equal its Curie temperature.