The force and the electric field between two point charges are given by: \[\vec
The energy density of the electromagnetic field is:
The energy density can be expressed from the potentials and currents as follows:
The wave equation \(\Box\Psi(\vec,t)=-f(\vec,t)\) has the general solution, with \(c=(\varepsilon_0\mu_0)^\): \
a derivation via multipole expansion will show that for the radiated energy, if \(d,\lambda\gg r\):
The energy density of the electromagnetic wave of a vibrating dipole at a large distance is:
The radiated energy can be derived from the Poynting vector \(\vec\): \(\vec=\vec\times\vec=cW\vec_v\). The irradiance is the time-averaged of the Poynting vector: \(I=\langle|\vec\,|\rangle_t\). The radiation pressure \(p_\) is given by \(p_=(1+R)|\vec\,|/c\), where \(R\) is the coefficient of reflection.
The wave equations in matter, with \(c_=(\varepsilon\mu)^\) the lightspeed in matter, are:
which after substitution of monochromatic plane waves: \(\vec=E\exp(i(\vec\cdot\vec-\omega t))\) and \(\vec=B\exp(i(\vec\cdot\vec-\omega t))\) yields the dispersion relation:
The first term arises from the displacement current, the second from the conductance current. If \(k\) is written in the form \(k:=k'+ik''\) it follows that:
This results in a damped wave: \(\vec=E\exp(-k''\vec\cdot\vec\,)\exp(i(k'\vec\cdot\vec-\omega t))\). If the material is a good conductor, the wave vanishes after approximately one wavelength, \(\displaystyle k=(1+i)\sqrt<\frac<\mu\omega>>\).
Because \(\displaystyle \frac<|\vec-\vec\,'|>=\frac\sum_0^\infty\left(\frac\right)^lP_l(\cos\theta)\) the potential can be written as: \(\displaystyle V=\frac<4\pi\varepsilon>\sum_n\frac\)
For the lowest-order terms this results in:
The continuity equation for charge is: \(\displaystyle\frac<\partial \rho><\partial t>+\nabla\cdot\vec=0\). The electric current is given by:
For most conductors: \(\vec=\vec/\rho\) holds, where \(\rho\) is the resistivity .
If the flux enclosed by a conductor changes this results in an induced voltage
If the current flowing through a conductor changes, this results in a self-inductance which opposes the original change: \(\displaystyle V_=-L\frac\). If a conductor encloses a flux then \(\Phi\): \(\Phi=LI\).
The magnetic induction within a coil is approximated by:
where \(l\) is the length, \(R\) the radius and \(N\) the number of coils. The energy contained within a coil is given by \(W=\frac LI^2\) and \(L=\mu N^2A/l\).
The capacitance is defined by:\(C=Q/V\). For a capacitor :
where \(d\) is the distance between the plates and \(A\) the surface of one plate. The electric field strength between the plates is \(E=\sigma/\varepsilon_0=Q/\varepsilon_0A\) where \(\sigma\) is the surface charge. The accumulated energy is given by \(W=\fracCV^2\). The current through a capacitor is given by \(\displaystyle I=- C\frac\).
For most PTC resistors \(R=R_0(1+\alpha T)\) holds approximately, where \(R_0=\rho l/A\). For a NTC: \(R(T)=C\exp(-B/T)\) where \(B\) and \(C\) depend only on the material.
If a current flows through a junction between wires of two different materials \(x\) and \(y\), the contact area will heat up or cool down, depending on the direction of the current: the Peltier effect . The generated or removed heat is given by: \(W=\Pi_It\). This effect can be amplified with semiconductors.
The thermal voltage between two metals is given by: \(V=\gamma(T-T_0)\). For a CuConstantan connection: \(\gamma\approx0.2-0.7\) mV/K.
In an electrical circuit with only stationary currents, Kirchhoff’s equations apply: for a closed loop : \(\sum I_n=0\), \(\sum V_n=\sum I_nR_n=0\).
If a dielectric material is placed in an electric or magnetic field, the field strength within and outside the material will change because the material will be polarized or magnetized. If the medium has an ellipsoidal shape and one of the principal axes is parallel with the external field \(\vec_0\) or \(\vec_0\) then the depolarizing fields are homogeneous.
\(\cal N\) is a constant depending only on the shape of the object placed in the field, with \(0\leq\leq1\). For a few limiting cases of ellipsoids the following holds: a thin plane: \(=1\), a long, thin bar: \(=0\), and a sphere: \(=\frac\).
The average electric displacement in a material which is inhomogenious on a mesoscopic scale is given by: \(\left\langle D \right\rangle=\left\langle \varepsilon E \right\rangle=\varepsilon^*\left\langle E \right\rangle\) where \(\displaystyle \varepsilon^*=\varepsilon_1\left(1-\frac<\phi_2(1-x)><\Phi(\varepsilon^*/\varepsilon_2)>\right)^\) and \(x=\varepsilon_1/\varepsilon_2\). For a sphere: \(\Phi=\frac+\fracx\). Further:
\[\left(\sum_i \frac<\phi_i>\right)^\leq\varepsilon^*\leq\sum_i \phi_i\varepsilon_i\]
This page titled 2: Electricity and Magnetism is shared under a CC BY license and was authored, remixed, and/or curated by Johan Wevers.
