 # Kirchhoff’s Second Law

Gustav Kirchhoff, a physicist from Germany, researched and found two laws concerning the electrical circuits involving lumped electrical elements. In the year 1845, he pursued the concepts of Ohm's law and Maxwell law and defined Kirchhoff’s first law (KCL) and Kirchhoff’s second law (KVL).

Kirchhoff’s current law or KCL is based on the law of conservation of charge. According to this, the input current to a node must be equal to the output current of the node. Further, the second law is discussed below in detail.

### State Kirchhoff’s Second Law

The second law by Kirchhoff is alternatively known as Kirchhoff’s voltage law (KVL). According to KVL, the sum of potential difference across a closed circuit must be equal to zero. Or, the electromotive force acting upon the nodes in a closed loop must be equal to the sum of potential difference found across this closed-loop.

Kirchhoff’s 2nd law also follows the law of conservation of energy, and this can be inferred from the following statements.

• In a closed-loop, the amount of charge gained is equal to the amount of energy it loses. This loss of energy is due to the resistors connected in this closed circuit.

• Also, the sum of voltage drops across the closed-circuit should be zero. Mathematically, it can be represented as ∑V=0.

### Limitation and Application of Kirchhoff’s Law

As per Kirchhoff, the law holds only in the absence of fluctuating magnetic fields in this circuit. So, it cannot be applied if there is a fluctuating magnetic field. Take a look at the applications of KVL.

• Sign Convention for KVL

Refer to this above image to find the signs of voltage when the direction of current in this loop is as shown.

• Kirchhoff’s Law Examples

Let us understand Kirchhoff’s voltage law with an example.

1. Take a closed-loop circuit or draw one as shown in the figure.

2. Draw the current flow direction in the circuit, and it might not be the actual direction of current flow.

3. At points A and B, I3 becomes the sum of I1 and I2. So, we can write I3 = I1 + I2.

4. According to Kirchhoff’s second law, the sum of potential drop in a closed circuit will be equal to the voltage. From this statement, we have

In loop 1: I1 * R1 + I3 * R3 = 10.

In loop 2: I2 * R2 + I3* R3 = 20.

In loop 3: 10 * I1 – 20 * I2 = 10 – 20.

By putting the value of R1, R2, and R3 in the above equations, we have

In loop 1: 10 I1+ 40 I3 = 10, or I1 + 4I3 = 1.

In loop 2: 20 I2+ 40 I3 = 20, or I2 + 2 I3 = 1.

In loop 3: 2 I2 –  I1 = 1.

1. According to Kirchhoff’s 1st law, we have I3 = I1 + I2. Substituting this in all 3 equations, we get

In loop 1: I1 + 4 (I1+I2) = 1, or 5 I1 + I2 = 1.      …………………(1)

In loop 2: I2 + 2 (I1+I2) = 1, or 2I1 + 3I2 = 1.       ……………….(2)

By equating equation 1 and 2, we have

5 I1 + I2 = 2I1 + 3I2, or 3 I1 = 2 I2

Therefore, I1 = -1/3 I2

By putting the value of I1 in loop 3 equation, we have

I1 = -0.143 A.

I2 = 0.429 A.

I3 = 0.286 A.

The above speculations and calculations prove that Kirchhoff’s voltage law holds true for these lumped electrical circuits.

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1. Kirchhoff’s Second Law is Based on Law of Conservation of Energy. Explain.

Ans. According to the law of conservation of energy, the electric energy gained within a closed electrical circuit is also lost within the loop. Also, the sum of potential difference in a closed circuit will be zero.

2. Explain the Sign Convention of Kirchhoff’s Second Law.

Ans. Electromotive force of a battery is considered to be positive when current flows from the negative terminal towards the positive terminal. If the current traverses across the resistor, then its potential difference is deemed to be negative.

3. What are Some Examples of Kirchhoff’s Voltage law?

Ans. Single loop circuits, complex electrical circuits, and charging circuits are examples of Kirchhoff’s law. You can see applications of this law in such closed-loop circuits.