Kirchhoffs Laws

Keywords Intramolecular kirchhoff laws Molecular electronics Molecular logic gates Single molecule electronic circuits Quantum hamiltonian computing... 

The Kirchhoff law is a direct consequence of the heat capacity at constant pressure being the derivative of enthalpy with respect to temperature. It is usually sufficient to assume that the heat capacity Cp is itself independent of temperature over the range of temperatures required, in which case Equation (3.19) simplifies to... 

Figure 3 shows a simple single-section RC low-pass filter network. KirchhofFs law tells us that... 

Fig. 7.84. Kirchhoffs law says that the sum of the inflowing currents at a junction must be equal to the sum of outflowing currents.

Absorption. The absorption coefficient, a(v), is related to the power due to spontaneous emission by KirchhofFs law,... 

The sum of the voltages over all transitions must be zero, in accordance with Kirchhoffs law. From this we get... 

The objective function (13) representing the total dissipation of kinetic energy of the flows at isothermal motion of fluid is proportional to the entropy production in the circuit and its transfer to the environment, i.e., proportional to the entropy accumulated by the isolated system (interconnection of the circuit and environment). The matrix equation (14) describes the first Kirchhoff law, which, as applied to hydraulic circuits, expresses the requirement for mass conservation. Equality (15) represents a balance between the energy generated and consumed in the circuit. 

Note that the coordinatewise optimization method has already found numerous practical applications to optimization of heat, oil, water, and gas supply systems (Merenkov and Khasilev, 1985 Merenkov et al., 1992 Sumarokov, 1976). As a matter of fact, in the algorithms used for applied problems the flow distribution was calculated not on the base of entropy maximization, but with the help of the closed system of equations of the first and second Kirchhoff laws. However, because of equivalence of approaches that are based on the principle of conservation and equilibrium (extremality) the Kirchhoff equations can be strictly replaced by thermodynamic relations. And the extreme thermodynamic approach in many cases should be preferable owing to the known low sensitivity of the extremal methods to variation of the space of variables. 

Fig. 8-4 Sketch showing model used tor deriving Kirchhoffs law.

Electric circuits or networks can be analyzed using both Ohm s law and KirchhofFs laws. For a circuit of resistors in series, as shown in Figure 2.6, the current flow in each resistor is the same (IR). 

Although the application of KirchhofFs laws offers basic tools to analyze a network, knowledge of certain network theorems, use of network equivalence, and use of reduction procedures simplify the process of network analysis. Basically, these theorems are applicable for linear networks. 

The Planck-Kirchhoff law allows a good approximation of the spectral radiance of any thermal radiator, the sources as well as the samples and detectors. Thermal radiators are characterized by a definite temperature as well as by their absorption coefficients f(i>) or a(i>), which describe the characteristic spectrum of the radiator ... 

Actually existing emitters are not black bodies. More realistic values may be calculated according to the Planck-Kirchhoff law, Eqs. 3.3-4 to 3.3-8. Table 3.4-4 shows typical molar decadic absorption coefficients for infrared absorption bands of character-... 

According to the Kirchhoff law, the capability of spectral emission equals the capability of spectral absorption. If the spectral absorptivities are known, it is therefore possible to determine vibrational temperatures from emission spectra. It is important to remember that the absorption coefficient throughout a band in a spectrum often varies considerably with the temperature. Temperatures are usually determined by fitting the calculated to the observed intensities. 

B Apply Kirchhoff law s law to determine the absorptivity of a surface when its emissivity is known,... 

For any pair of non-adjacent vertices, the resistance distance is the effective resistance calculated according to the two classical Kirchhoff laws for series and parallel electrical circuits some examples of calculations are given in Box R-1 for ethylbenzene. 

Atomic absorp- Elemental analysis method based on Kirchhoff law. According tion spectros- to this law, all atoms are able to absorb light quanta with the copy same wavelength which they are able to emit. 

Gx is usually called the irradiation of an enclosure or the incident radiation on a body. For an ideal absorber (black body), a . = 1, and the Kirchhoff law,... 

Also, in terms of the (total) emissivity e, the Kirchhoff law becomes... 

Naturally, electrical engineers have designed equivalent circuits for nonelectrical wave phenomena. The waves may or may not be confined to cables. For simple propagating waves, the equivalent circuits are often called transmission line models. The transmission line has two ports representing input and output. The input-output relation can be predicted by applying the Kirchhoff laws to the set of elements located in between. The circuit elements may be simple resistors or capacitors, but their electrical impedance may also be a more complicated function of frequency (see, for instance. Fig. 6)... 

Figure 6a shows the transmission hne representing a viscoelastic layer [64]. Every layer is represented by a T . The apphcation of the Kirchhoff laws to the Ts reproduces the wave equation and the continuity of stress and strain. The detailed proof is provided in [4]. To the left and to the right of the circuit are open interfaces (ports). These can be exposed to external shear waves. They can also be connected to the ports of neighboring layers (Fig. 6b). Alternatively, they may just be short-circuited, in case there is no stress acting on this surface (left-hand side in Fig. 6c). Finally, if the stress-speed ratio Zl (the load impedance, see below) of the sample is known, the port can be short-circuited across an element of the form AZl, where A is the active area (right-hand side in Fig. 6c). Figure 6c shows a viscoelastic layer which is also piezoelectric. This equivalent circuit was first derived by Mason [4,55]. We term it the Mason circuit. The capacitance, Co, is the electric capacitance between the electrodes. The port to the right-hand side of the transformer is the electrical port. The series resonance frequency is given by the condition that the impedance of the acoustic part (the stress-speed ratio, aju) be zero, where the acoustic part comprises all elements connected to the left-hand side of the transformer.

When applying the Kirchhoff laws to such a network, one finds the same resonance conditions as with the mathematical and the optical approaches. Why should one bother going through these transformations if the results are the same There are important benefits tied to the use of equivalent circuits ... 

All acoustic, electric, and piezoelectric parameters can be displayed in a single graph, which is not just a cartoon. The graph, in conjunction with the Kirchhoff laws, predicts the behavior of the resonator. Anyone who can... 

KIRCHHOFF S LAWitAn important generalization concerning the radiating power of a substance is KirchhofFs law, which states that, at temperature equilibrium, the ratio of the total radiating power of any body to its absorptivity depends only upon the temperature of the body. Thus, consider any two bodies in temperature equilibrium with common surroundings. Kirchhoffs law states that... 

Thus, when any body is at temperature equilibrium with its surroundings, its emissivity and absorptivity are equal. This relationship may be taken as another statement of Kirchhoffs law. In general, except for blackbodies or gray bodies, absorptivity and emissivity are not equal if the body is not in thermal equilibrium with its surroundings. 

Kirchhoffs law (Eq. 7.27) may be used to replace the absorptivity in these relations with emissivity if the restrictions of Table 7.1 are observed. Thus, data on one of the radiative properties can often be used to generate the others, although care must be used to avoid violating the restrictions of Table 7.1. 

The values iv are calculated according to KirchhofFs laws. At the nodes the equation... 

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