**Question:**Kirchhoff's Laws - Current Law & Voltage Law

**Answer:**In 1845, German physicist Gustav Kirchhoff first described two laws that became central to electrical engineering. The laws were generalized from the work of Georg Ohm. The laws can also be derived from Maxwell’s equations, but were developed prior to Maxwell’s work.

The following descriptions of Kirchhoff's Laws assume a constant current. For time-varying current, or alternating current, the laws must be applied in a more precise method.

### Kirchhoff's Current Law

Kirchhoff's Current Law, also known as Kirchhoff's Junction Law and Kirchhoff's First Law, defines the way that electrical current is distributed when it crosses through a junction - a point where three or more conductors meet. Specifically, the law states that:Since current is the flow of electrons through a conductor, it cannot build up at a junction, meaning that current is conserved: what comes in must come out. When performing calculations, current flowing into and out of the junction typically have opposite signs. This allows Kirchhoff's Current Law to be restated as:The algebraic sum of current into any junction is zero.

The sum of current into a junction equals the sum of current out of the junction.

### Kirchhoff's Current Law in action

In the picture to the right, a junction of four conductors (i.e. wires) is shown. The currents*i*

_{2}and

*i*

_{3}are flowing into the junction, while

*i*

_{1}and

*i*

_{4}flow out of it. In this example, Kirchhoff's Junction Rule yields the following equation:

i_{2}+i_{3}=i_{1}+i_{4}

### Kirchhoff's Voltage Law

Kirchhoff's Voltage Law describes the distribution of voltage within a loop, or closed conducting path, of an electrical circuit. Specifically, Kirchhoff's Voltage Law states that:The voltage differences include those associated with electromagnetic fields (emfs) and resistive elements, such as resistors, power sources (i.e. batteries) or devices (i.e. lamps, televisions, blenders, etc.) plugged into the circuit.The algebraic sum of the voltage (potential) differences in any loop must equal zero.

Kirchhoff's Voltage Law comes about because the electrostatic field within an electric circuit is a conservative force field. As you go around a loop, when you arrive at the starting point has the same potential as it did when you began, so any increases and decreases along the loop have to cancel out for a total change of 0. If it didn't, then the potential at the start/end point would have two different values.

### Positive and Negative Signs in Kirchhoff's Voltage Law

Using the Voltage Rule requires some sign conventions, which aren't necessarily as clear as those in the Current Rule. You choose a direction (clockwise or counter-clockwise) to go along the loop.When travelling from positive to negative (+ to -) in an emf (power source) the voltage drops, so the value is negative. When going from negative to positive (- to +) the voltage goes up, so the value is positive.

When crossing a resistor, the voltage change is determined by the formula *I***R*, where *I* is the value of the current and *R* is the resistance of the resistor. Crossing in the same direction as the current means the voltage goes down, so its value is negative. When crossing a resistor in the direction opposite the current, the voltage value is positive (the voltage is increasing).

### Kirchhoff's Voltage Law in action

If you click on the picture to the right, you can advance to a second picture that depicts a loop*abcd*. If you begin at

*a*and advance clockwise along the interior loop, the Voltage Law yields the equation:

In this case, the current will also be clockwise. Crossing the resistors will result inv_{1}+v_{2}+v_{3}+v_{4}= 0

*v*

_{1},

*v*

_{2}, and

*v*

_{3}all being negative. Since you're crossing from negative to positive,

*v*

_{4}will be positive. If you consider the dotted line that has the

*R*

_{5}resistor, you get a total of three loops in the circuit. The first one has already been described. One loop is the largest loop and another is the smallest loop at the bottom, to yield the equations:

The second equation is of special interest, since it indicates thatv_{1}+v_{2}+v_{5}+v_{4}= 0 (abcdtaking the new path instead ofR_{5})v_{3}+v_{5}= 0 (the small loopcd)

*v*

_{3}= -

*v*

_{5}. This makes sense, because both currents will be travelling from

*c*to

*d*, so on the small loop you'll cross one resistor with the current and the other resistor against the current. If the resistors are of equal value, then the current in both paths will be equal.

### Bringing It All Together

In the voltage loop diagram, we see that at junction*c*there are three conductors. Current enters from the top, then goes out in the other two directions and, from the Current Law, we know the algebraic sum of these must be zero. If we knew the resistance values of the resistors, and the current coming into the junction, we could use the Voltage Law equations to determine the currents in each of the lower paths, if the resistors were unequal.

This is why Kirchhoff's Laws are such powerful tools.