Kirchhoff Voltage Law Explanation: Derivation/Proof

[sambarbarian]

hi! i understand how the krichoff current law works , but can someone explain me the derivation/proof of the voltage law .

 

[tenchotomic]

It follows directly from law of conservation of energy.I haven't seen it's derivation anywhere.

 

[sambarbarian]

does anyone else have a mathematical or logical proof ?

 

[vanhees71]

The Kirchhoff rules are special cases of Maxwell's equations for quasistationary problems. For AC it is simple to state: It's applicable as long as the geometrical extensions of the electric circuit are small compared to the wavelength of the electromagnetic fields, i.e., if $l \ll \lambda=c/f$, where $f$ is the frequency of your AC, because then the retardation of electromagnetic fields can be neglected. You can find a derivation in the marvelous (however somewhat a bit older) textbook

A. Sommerfeld, Lectures on Theoretical Physics, Vol. 3 (Electromagnetism).

 

[Muphrid]

It should follow conceptually from the idea that the voltage at any given point has only one value. If you go around a loop and come back to that point, the voltage should not be different--i.e. the sum of all voltage changes around that closed loop therefore must be zero.

 

[mikelepore]

Muphrid is right. It it true of any situation in which each point in space has a particular value. It would even be true of a chart of random numbers. If you travel in any closed path, the sum of all positive transitions and negative transitions must be zero.

But why does each point in space have a unique value? It is because the electric force is a conservative force. The work that an external agent does on a charge to make it go from point A to point B is equal in magnitude and opposite in sign to the work that the electric field does on the charge to make it go from point B to point A. In other words, such a thing as potential exists in the first place.

 

[sambarbarian]

muphrid and mikelepore , together your comments answered my question ... ty

 

[Studiot]

I know it is common these days to be clever and fashionable and teach Kirchoff's voltage law as an equation summing to zero.

However that was not the original formulation and I would not recommend it as it can actually lead to incorrect results if used in that way.

For instance connect a 12 volt battery in parallel with a 6 volt battery and a load and try performing a KVL analysis.

Or try to perform a KVL on a circuit containing a current source.

The original formulations was in line with the proinciple of conservation of energy as mentioned earlier.

The sum of the EMFs equals the algebraic sum of the IR products in a closed loop

Used in this form KVL will not betray you.

 

[vanhees71]

Even better is to remember that Kirchhoff's Law is an integrated form of Faraday's Law stating

$$\vec{\nabla} \times \vec{E} = -\partial_t \vec{B}.$$

This is integrated along the wires and compact resistors, capacitors, and inductances. This also explains the sign of currents and emf's as being defined according to the right-hand rule, implemented in the definition of the rotation (curl) operator on the left-hand side via Stoke's theorem for infinitesimal surfaces and their boundaries, where by convention the relative orientation of the area-normal vectors and the boundary curve is according to the right-hand rule.

 

[Studiot]

Are you sure Kirchoff knew all that?

 

[vanhees71]

I'm not so sure historically. Of course, we don't need to bother know what as been known then. The good thing with natural sciences is that you don't need to learn too much of outdated stuff but you can start with the best knowledge one has, and in the context of this topic that's Maxwell's electrodynamics.

 

[Studiot]

Yes if you are at University, but I think Sam is in high school,although he is asking some penetrating questions.

 

[sambarbarian]

Yes if you are at University, but I think Sam is in high school,although he is asking some penetrating questions.

yes I am still in high school

 

[Studiot]

Not 'still', Sam, you are doing well.

Keep questioning as you have done and you will go far.

 

[mikelepore]

For instance connect a 12 volt battery in parallel with a 6 volt battery and a load and try performing a KVL analysis.

When designing with ideal circuit elements, that connection is undefined, like a math lesson about never dividing by zero. You placed 12-6= 6 volts across the zero resistance of the interconnecting wires, and power v^2/R went to infinity. But does anyone know what happens if you do it in real lfe? Do the wires get hot and perhaps melt? Or do the chemical reactions in the batteries fail to deliver the advertised voltage?

 

[vanhees71]

I didn't know that "sambarbarian" is at high school, and for sure I can only also encourage you to ask such questions. These are really the right ones to ask concerning science! I really thought you are at undergrad university level, and of course I do not expect that you know Maxwell's equations at high school! Sorry for my misunderstanding.