GATE Electromagnetics formula sheet

Force on $Q_2$ due to $Q_1$; ${\hat{a}}_{12}$ is the unit vector from $Q_1$ toward $Q_2$ and $R=|{\mathbf{r}}_2-{\mathbf{r}}_1|$. A positive product $Q_1{Q}_2$ means repulsion. Valid for point charges (or bodies small compared with $R$). In a homogeneous dielectric filling all space replace ${\epsilon }_0$ by $\epsilon$: the force falls by the factor ${\epsilon }_r$.

Field at distance $R$ from a point charge $Q$: away from a positive charge, toward a negative one. A charge $q$ placed in the field feels $\mathbf{F}=q\mathbf{E}$. Units: V/m, the same as N/C.

Linear, isotropic medium. $\mathbf{M}$ is the magnetization (magnetic dipole moment per unit volume, A/m) and ${\chi }_m$ the magnetic susceptibility. In a ferromagnetic material $\mu$ is not a constant: it depends on $H$ and on the magnetic history (see the B-H curve below).

$d\mathbf{S}$ is normal to the surface. For a uniform $B$ normal to a flat area $A$, $\Phi =BA$.

$\Psi =N\Phi$ is the flux linkage of $N$ tightly wound turns. Holds for any closed circuit of thin wire, stationary or moving, including a bar sliding on rails. Where the current path through a moving conductor is not a definite wire (a rotating disc), compute the emf from the $\mathbf{u}\times \mathbf{B}$ integral below instead. A positive emf drives current in the sense that is right-handed about the $d\mathbf{S}$ used to compute $\Phi$.

Path fixed in space, $\mathbf{B}$ changing with time.

In rad/m; $\lambda$ is the wavelength in the medium.

${\lambda }_0=c/f$ is the free-space wavelength. Refractive index $n=\sqrt{{\mu }_r{\epsilon }_r}=c/v_p$. The frequency does not change when a wave enters a new medium; the wavelength does.

Sinusoidal steady state with $e^{j\omega t}$; the third equation follows by differentiating the first and substituting the second.

$V_0^{+}$ is the wave travelling toward the load and $V_0^{-}$ the reflected wave. The minus sign in $I(z)$ matters: the reflected wave's current flows back toward the generator.

$k_c$ is the mode's cut-off wavenumber, set by the guide's shape and size, and $f_c=k_{c}u/(2\pi )$. $\beta$ is real (the mode propagates) only when $f>f_c$.

Same formula for TE and TM. Allowed indices: ${\mathrm{TE}}_{mn}$ with $m,n=0,1,2,\dots$ but not both zero, and ${\mathrm{TM}}_{mn}$ with $m,n\ge 1$. A filling with ${\mu }_r=1$ divides every air-filled cut-off by $\sqrt{{\epsilon }_r}$.

$D$ = largest dimension of the antenna (a length here, not the directivity). At $r=2D^2/\lambda$ the path-length difference between the centre and the edge of the antenna is $\lambda /16$ (phase error $\pi /8$). For an antenna large compared with the wavelength, the reactive near field extends to about $r=0.62\sqrt{D^3/\lambda }$ and the radiating near field from there to $2D^2/\lambda$. For an electrically small antenna ($D\ll \lambda$) the binding condition is $r\gg \lambda$; the radiation and induction terms of a Hertzian dipole are equal in magnitude at $r=\lambda /(2\pi )$.

The middle form holds in the far field only, where $E$ and $H$ are in phase and $|E|=\eta |H|$. $E$ is a peak phasor. In a lossless medium $P_{\text{rad}}$ is the same through every closed surface that encloses the antenna.