Kirchhoff\\\'s_circuit_laws

For other laws named after Gustav Kirchoff, see Kirchhoff\'s laws. Not to be confused with Kerckhoffs\' principle.

Kirchhoff\'s circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. Widely used in electrical engineering, they are also called Kirchhoff\'s rules or simply Kirchhoff\'s laws (see also Kirchhoff\'s laws for other meanings of that term).

Both circuit rules can be directly derived from Maxwell\'s equations, but Kirchhoff preceded Maxwell and instead generalized work by Georg Ohm.

Kirchhoff\'s Current Law (KCL)

The current entering any junction is equal to the current leaving that junction. i₁ + i₄ = i₂ + i₃

This law is also called Kirchhoff\'s first law, Kirchhoff\'s point rule, Kirchhoff\'s junction rule (or nodal rule), and Kirchhoff\'s first rule.

The principle of conservation of electric charge implies that:

At any point in an electrical circuit where charge density is not changing in time, the sum of currents flowing towards that point is equal to the sum of currents flowing away from that point.

A charge density changing in time would mean the accumulation of a net positive or negative charge, which typically cannot happen to any significant degree because of the strength of electrostatic forces: the charge buildup would cause repulsive forces to disperse the charges.

However, a charge build up can occur in a capacitor, where the charge is typically spread over wide parallel plates, with a physical break in the circuit that prevents the positive and negative charge accumulations over the two plates from coming together and cancelling. In this case, the sum of the currents flowing into one plate of the capacitor is not zero, but rather is equal to the rate of charge accumulation. However, if the displacement current dD/dt is included, Kirchhoff\'s current law once again holds. (This is only required if one wants to apply the current law within the capacitor. In circuit analyses, however, the capacitor as a whole is typically treated as a unit, in which case the ordinary current law holds since the net charge is always zero.)

More technically, Kirchhoff\'s current law can be found by taking the divergence of Ampère\'s law with Maxwell\'s correction and combining with Gauss\'s law, yielding:

$\backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}\; =\; -\backslash nabla\; \backslash cdot\; \backslash frac\{\backslash partial\; \backslash mathbf\{D\}\}\{\backslash partial\; t\}\; =\; -\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}$

This is simply the charge conservation equation (in integral form, it says that the current flowing out of a closed surface is equal to the rate of loss of charge within the enclosed volume). Kirchhoff\'s current law is equivalent to the statement that the divergence of the current is zero, true for time-invariant ρ, or always true if the displacement current is included with J.

A matrix version of Kirchhoff\'s current law is the basis of most circuit simulation software, such as SPICE.

Kirchhoff\'s Voltage Law (KVL)

The sum of all the voltages around the loop is equal to zero. v₁ + v₂ + v₃ + v₄ = 0

This law is also called Kirchhoff\'s second law, Kirchhoff\'s loop(or mesh) rule, and Kirchhoff\'s second rule.

The directed sum of the electrical potential differences around any closed circuit must be zero.

This statement is equivalent to the statement that a single-valued electric potential can be assigned to each point in the circuit (in the same way that any conservative vector field can be represented as the gradient of a scalar potential).

(This could be viewed as a consequence of the principle of conservation of energy. Otherwise, it would be possible to build a perpetual motion machine that passed a current in a circle around the circuit.)

Considering that electric potential is defined as a line integral over an electric field, Kirchhoff\'s voltage law can be expressed equivalently as

$\backslash oint\_C\; \backslash mathbf\{E\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; 0,$

which states that the line integral of the electric field around closed loop C is zero.

or

$\backslash sum\; \{IR\}\; +\; \backslash sum\; \backslash mathcal\{E\}\; =\; 0\backslash ,$ (round a loop/mesh)

This is a simplification of Faraday\'s law of induction for the special case where there is no fluctuating magnetic field linking the closed loop. In the presence of a changing magnetic field the electric field is not conservative and it cannot therefore define a pure scalar potential—the line integral of the electric field around the circuit is not zero. This is because energy is being transferred from the magnetic field to the current (or vice versa). In order to "fix" Kirchhoff\'s voltage law for circuits containing inductors, an effective potential drop, or electromotive force (emf), is associated with each inductance of the circuit, exactly equal to the amount by which the line integral of the electric field is not zero by Faraday\'s law of induction.

See also

• Kirchhoff\'s law of thermal radiation
• Faraday\'s law of induction
• Gauss\'s law
• Maxwell\'s equations
• Conservation of energy
• SPICE (Simulation Program with Integrated Circuits Emphasis)

KVL may also be stated as " the algebraic sum of various potential drops across an electrical circuit is equal to the electromotive force acting on the circuit"

References

• Paul, Clayton R. (2001). Fundamentals of Electric Circuit Analysis. John Wiley & Sons. ISBN 0-471-37195-5. 
• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7. 
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8. 

External links

• http://www.sasked.gov.sk.ca/docs/physics/u3c13phy.html

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