8. Electric currentSavchenko Formulas, chapter 8 of 14, 23 formulas
Sections follow the book, and within a section the most used formulas come first. An italic problem number means the formula appears in its statement. Rest the cursor on a number to see the statement.
$\displaystyle I = \frac{qv}{l}\ \text{(заряд }q\text{, пробегающий контур длины }l\text{ со скоростью }v)$$\displaystyle I = \frac{e\omega}{2\pi}\ \text{(электрон на орбите)}$
I
current
$dq$
charge crossing a section in time $dt$
Current is the charge passing through a cross-section of a conductor per unit time. A charge going round a ring makes an average current $q/T$, and a charge flying between plates a current $qv/d$ in the wire joining them. For a steady current the charge passed is $It$, otherwise the integral.
$\displaystyle \rho = \frac{j}{v}\ \text{(плотность заряда в пучке)}$$\displaystyle n = \frac{\rho N_A}{M}\ \text{(один электрон на атом)}$
j
current density
n
carrier number density
v
their mean drift velocity
S
cross-section area
Through an area $S$ each second pass the carriers of a cylinder of length $v$,$nSv$ of them with charge $e$ each. In a metal about one free electron per atom gives $n \sim 10^{29}$ m$^{-3}$ and a drift speed of a fraction of a millimetre per second at ordinary currents. In a beam the charge density is $j/v$, so charge piles up where the particles are slower.
$\displaystyle j = \frac{4\varepsilon_0}{9}\sqrt{\frac{2e}{m}}\,\frac{U^{3/2}}{d^2}\ \text{(закон трёх вторых)}$
$\varphi$
potential between cathode and anode
$\rho$
charge density of the electrons
v
their speed, from $mv^2/2 = e\varphi$
U
anode voltage
In a vacuum diode the electrons make their own field, and the current follows from three equations together, Poisson's equation for the potential, $j = \rho v$ and energy conservation for the speed. The solution is a power law, $\varphi \propto x^{4/3}$, and the current grows as $U^{3/2}$ while the cathode supplies enough electrons. By similarity, raising the voltage $n$ times raises the current $n^{3/2}$ times for any electrode geometry.
field at the charged surface, $\sigma/\varepsilon_0$
A moving charge is a current, and through any fixed section per second passes the charge lying on a length $v$. A belt with surface density $\sigma$ carries a current $\sigma v b$, a rotating disc with charge $q$ a current $q\omega/2\pi$. The charge density on a metal belt comes from the field at its surface, $\sigma = \varepsilon_0 E$, and a dielectric's surface in a capacitor carries the bound charge $\varepsilon_0(\varepsilon-1)E$.
In a current-carrying conductor the field is not zero but proportional to the current density, with a coefficient that is a property of the material. In a steady current $\operatorname{div}\vec j = 0$, and with $\vec j = \lambda\vec E$ this makes the current distribution in bulk conductors an electrostatics problem. The voltage between points is $\int \vec E\cdot d\vec l$ along any path inside the conductor. In the Drude model the conductivity is expressed through the electrons' mean free time.
$\displaystyle R = \frac{1}{4\pi\lambda}\left(\frac{1}{r_1} - \frac{1}{r_2}\right)\ \text{(между концентрическими сферами)}$$\displaystyle R = R_0[1 + \alpha(T - T_0)]\ \text{(температурная зависимость)}$
$\rho$
resistivity
l
length of the conductor
S
cross-section area
$\alpha$
temperature coefficient of resistance
Resistance grows with length and falls with cross-section, since at one current the field $j/\lambda$ accumulates along the length while the current density $I/S$ falls with the area. A conductor of varying section is cut into layers whose resistances add, which gives the resistance between spheres or coaxial cylinders. Stretching a wire at constant volume raises the resistance as the square of the length, and in metals it grows with temperature.
A steady current stores no charge, so the normal component of $j$ is continuous while the field $j/\lambda$ differs on the two sides. The jump of $E_n$ means a surface charge $\varepsilon_0(E_{2n} - E_{1n})$ at the boundary, and a similar volume charge appears wherever the conductivity changes along the current. The tangential component of $E$ is continuous, hence the refraction of current lines with tangents in the ratio of the conductivities.
strength of the non-electrostatic force per unit charge
$A_{\text{стор}}$
work of the non-electrostatic forces carrying a charge $q$ round the circuit
The electrostatic field is conservative and cannot drive a current round a closed loop, that takes non-electrostatic forces, chemical, magnetic or inertial, and their work per unit charge is the emf. A galvanic cell's emf equals the reaction energy per charge transferred, a spun-up conductor's is set by the inertial force on an electron. The power delivered by a source is $\mathcal{E}I$.
Charge in a weakly conducting medium leaks away with a current $j = E/\rho$, while the field is made by that charge itself, $E = Q/4\pi\varepsilon\varepsilon_0 r^2$ for a ball, so the current $Q/\varepsilon\varepsilon_0\rho$ does not depend on size and the charge decays exponentially. For any capacitor filled with such a medium the product $RC$ equals $\varepsilon\varepsilon_0\rho$, since $R$ and $C$ follow from one geometry.
$$F v = I^2 R, \qquad I = \frac{qv}{l}, \qquad \varphi = IR$$
F
force moving the charged body along the conductor
v
its steady speed
q
charge of the body
l
length of the loop
R
resistance of the loop
If a charged body is dragged along a closed conductor, a current $qv/l$ flows in it and all the force's work becomes Joule heat. Equating the mechanical power $Fv$ to the electrical $I^2R$ gives the steady speed and the voltage across the ring. It is a model of a current source whose driving force comes from an outside body.
Each ion discharged at an electrode carries $ze$ of charge, so the number of moles deposited is proportional to the charge passed. A mole of a monovalent substance needs a charge $F$. This also measures the electron's charge, given Avogadro's number.
The current through a uniform segment is proportional to the applied voltage, and the same relation gives the voltage across an element carrying a known current. With one current, voltages across series segments are as the resistances, and across parallel ones the currents are inversely proportional to the resistances. A voltmeter of resistance $R_V$ reads $IR_V$ and passes a current of its own.
currents in the branches meeting at a node or forming a loop
Rk
resistances of the loop's branches
$\mathcal{E}_k$
emfs in the loop, signed by the direction of traversal
Charge does not pile up at a node, so the currents flowing in add up to those flowing out. Around any closed loop the voltage drops $IR$ add up to the emfs, since the potential returns to its value. The number of independent equations equals the number of unknown currents, and in symmetric networks nodes at equal potentials are joined to simplify the net.
$\displaystyle w = \vec j\cdot\vec E = \lambda E^2\ \text{(мощность в единице объёма)}$$\displaystyle P_{\text{ист}} = \mathcal{E}I, \qquad P_{\text{внутр}} = I^2 r$
P
power dissipated in the segment
I
current
U
voltage across the segment
Q
heat released in time $t$
In a time $t$ a charge $It$ passes through the segment and the field does work $UIt$ on it, all of which becomes heat in a resistor. A source delivers $\mathcal{E}I$, of which $I^2r$ heats the source itself. One picks whichever of the three forms uses the known quantity, in series the heat splits in proportion to the resistances, in parallel inversely. In a bulk conductor the power per unit volume is $\lambda E^2$.
In series the current is common and the voltages add, in parallel the voltage is common and the currents add. A complicated network is simplified by joining points of equal potential and dropping branches without current, and an infinite chain is solved by noting that adding one more link does not change it. A series resistor on a voltmeter and a shunt on an ammeter extend the instruments' ranges.
The current in a closed circuit is set by the emf and the sum of all resistances, the source's internal resistance included. The terminal voltage is below the emf by the drop $Ir$ inside the source and equals the emf only with no load, while with current driven inward it exceeds the emf. Several sources in one loop add their emfs with signs, and the internal resistance follows from two measurements of current and voltage.
The load power $I^2R$ with $I = \mathcal{E}/(R+r)$ peaks at $R = r$, when half the source's power heats the source itself and the efficiency is one half. For $R \gg r$ the efficiency approaches one but the power is small. Two loads giving the same power are related by $R_1R_2 = r^2$.
$\displaystyle I^2 R = \sigma S T^4\ \text{(нить лампы, излучение)}$$\displaystyle I^2Rt = cm\,\Delta T\ \text{(нагрев без отвода тепла)}$
$\kappa$
heat-loss coefficient, power per degree
T0
ambient temperature
$\alpha$
temperature coefficient of resistance
A conductor settles at the temperature where the Joule heat equals the heat carried away, proportional to $T - T_0$ by conduction and convection, as $T^4$ by radiation. Since a metal's resistance grows on heating, at a fixed current the dissipation grows with temperature, and if $\kappa < I^2R_0\alpha$ there is no equilibrium and the conductor burns out. Without heat loss the warming follows $cm\,\Delta T$.
$\displaystyle q = q_0 e^{-t/RC}, \qquad I = \frac{\mathcal{E}}{R}e^{-t/RC}$$\displaystyle t = RC\ln\frac{U_0}{U}\ \text{(время разрядки до }U)$
q
charge of the capacitor
$\mathcal{E}$
emf of the source
$\tau$
time constant $RC$
The current is $(\mathcal{E} - q/C)/R$ and falls as the capacitor charges, so the charge approaches $C\mathcal{E}$ exponentially with time constant $RC$. On discharge the current is proportional to the remaining charge and decays by the same exponential. In a time $RC$ the quantities change by a factor $e$, and after a few time constants the process is essentially over.
current through the capacitor's branch in the steady state
UC
voltage on the capacitor, the potential difference of the nodes it joins
Once the currents have settled no current flows through the capacitor, so its branch can be dropped, the node potentials found from Ohm's and Kirchhoff's laws, and then the capacitor restored with the voltage between its nodes. The charge that passed through the source on switching equals the change of the capacitor's charge, and the heat comes from the energy balance. If a capacitor is charged and discharged periodically, the mean currents follow from the charge moved per cycle.
$\displaystyle Q = \frac{C(\mathcal{E} - U_0)^2}{2}\ \text{(дозарядка от }U_0\text{ до }\mathcal{E})$$\displaystyle Q_1 : Q_2 = R_1 : R_2\ \text{(тепло в последовательных резисторах)}$
$A_{\text{ист}}$
work of the source, emf times the charge passed
W
energy stored in the capacitor
Q
heat in the wires, whatever the resistance
A source of constant emf does work $\mathcal{E}q$, but the capacitor stores only half, $q^2/2C$, and the other half becomes heat regardless of the circuit's resistance, since at small $R$ the current is larger but the time shorter. Charging in steps, by small increments of voltage, cuts the loss, to zero in the limit. The heat between series resistors splits in proportion to the resistances, the current being the same.
$$I = \alpha U^2, \qquad I = \frac{\mathcal{E} - U}{R}$$
I
current through the element
U
voltage across it
$\alpha$
coefficient of the characteristic
R
series resistance
For an element with a known current-voltage characteristic Ohm's law is replaced by the characteristic $I(U)$ itself, and the rest of the circuit supplies a second relation between the same $I$ and $U$, usually linear. Their intersection, by algebra or on a graph, is the operating point. This handles a lamp with $I \propto U^2$ or a diode in series with a resistor.
A charge between connected plates induces on them charges that depend on its position, inversely to the distances, since the plates share one potential. As the charge moves the induced charges flow through the wire, and the current is $qv/d$ throughout the flight, not only at impact. The same gives the circuit current from an electron crossing a vacuum tube.