[Physics Class Notes] on Kirchhoff’s Second Law Pdf for Exam

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 differences 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 differences 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 [sum 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 the 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. 

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

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

At points A and B, [I_{3}] becomes the sum of [ I_{1} and I_{2} ]. So, we can write [I_{3} = I_{1} + I_{2} ]. 

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

In loop 1: [I_{1} * R_{1} + I_{3} * R_{3} = 10]. 

In loop 2: [ I_{2} * R_{2} + I_{3}* R3 = 20 ]. 

In loop 3: [ 10 * I_{1} – 20 * I_{2} = 10 – 20 ]. 

By putting the value of [ R_{1}, R_{2}, and R_{3} ] in the above equations, we have 

In loop 1: [ 10 I_{1}+ 40 I_{3} = 10, or I_{1} + 4 I_{3} = 1 ]. 

In loop 2: [20 I_{2} + 40 I_{3} = 20, or I_{2} + 2 I_{3} = 1]. 

In loop 3: [ 2 I_{2} – I_{1} = 1]. 

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

In loop 1: [I_{1} + 4 (I_{2} + I_{2}) = 1, or 5 I_{1} + I_{2} = 1]…………………(1)

In loop 2: [I_{2} + 2 (I_{1} + I_{2}) = 1, or 2I_{1} + 3I_{2} = 1]……………….(2)

By equating equation 1 and 2, we have 

[5 I_{1} + I_{2} = 2I_{1} + 3I_{2}, or 3 I_{1} = 2 I_{2}]

Therefore, [I_{1} = -1/3 I_{2}]

By putting the value of [I_{1}] in loop 3 equation, we have 

[I_{1}] = -0.143 A. 

[I_{2}] = 0.429 A. 

[I_{3}] = 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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