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# Kirchhoff's Voltage Law - Circuit Diagrams, Practice problems, FAQs

You go to your physics lab and find 3 resistors and 2 cells lying around. You can make a combination of circuits using these with connecting wires. One such circuit diagram which can be constructed would look like this

Now if one were to ask what is the current flowing through this circuit, you would apply Ohm’s law and find the net resistance of the circuit, and current is nothing but the ratio of net emf to the net resistance. But the problem becomes complex if the circuit is divided into many branches as shown in the given figure.

Gustav Robert Kirchoff, born on March 12, 1824 in Konigsberg, Russia researched a lot on electricity. This made him come up with Kirchoff’s laws in 1845 which can be used for finding both current and voltage difference. Apart from electricity, Kirchoff is also known for making contributions to black body radiation and spectroscopy. In this article, we will explore Kirchhoff's Voltage law in detail.

• Kirchoff’s current law
• Definition of Kirchoff’s voltage law (KVL)
• Sign convention for the application of Kirchoff’s laws
• Practice problems
• FAQs

## Kirchoff’s current law

It says that the algebraic sum of all the currents into a junction is equal to zero.

Incoming current= outgoing current.

Δ

Since current I is nothing but the flow of charges in unit time.

$\Delta Q={\Delta Q}_{1}+{\Delta Q}_{2}.$

Hence, KCL is in accordance with the law of conservation of charge.

## Definition of Kirchoff’s voltage law (KVL)

Kirchoff’s voltage law, also known as Kirchoff’s loop rule, states that the algebraic sum of the emfs of the battery in a closed circuit is equal to the algebraic sum of the products of the resistances and currents which are in the circuit.

In other words, the sum of all the potential differences associated with the cell emfs and those of corresponding resistive elements in a loop is zero.

i.e., ⅀ΔV = 0.

## Sign convention for the application of Kirchoff’s laws

Current I flows between A and B and a resistance R is connected between the two points.

While traversing left to right, the direction of flow of current is also from left to right. When the potential difference between A and B reduces in the same direction as the flow of current, i.e from left to right, the product of current and resistance is negative i.e -IR.

While traversing in the opposite direction i.e., right to left while the direction of current flow is from left to right, the product of current and resistance is positive i.e +IR.

In the following figure, i is the current flowing in the circuit. Going in the clockwise direction along ABCDA, KVL would give us

Conversely, going in the anticlockwise direction around ADCBA, we get

Equations (i) and (ii) are the same. The decrease in emf is balanced by the product of current and resistance, conserving voltage in the closed loop. KVL is a consequence of conservation of energy.

Video explanation

## Practice problems

Q. In the given circuit, find the currents i1 and i2.

(a) 1 A , 2 A        (b)          (c)           (d) 5 A , 2 A

A. c

Let i1 be the current coming out of the 20 V battery in the left branch and i2 be the current coming of 20 V battery in the right branch. Applying KCL at junction C we get,

is the current flowing through the middle branch.

Applying KVL in the loop ABCFA, we get

Applying KVL in the loop CDEFC,

Solving (iii) and (iv), we get

Q. Calculate the currents in the different parts of the circuit

A. The respective currents through the resistors are shown in the figure. Applying KCL at junction B, we get

i= i+ i3--(i)

Applying Kirchoff’s second law in loop 1,

-4i+ 4 - 2i+ 2 = 0--(ii)

Applying Kirchoff’s second law in loop 2,

-2i3-6-4i3-4=0--(iii)

Solving equations (i), (ii) and (iii) we get

i1=1 A,

and

Q. Find the potential difference between points A and B in the figure shown below.

(a)+9 V                (b)-V                 (c) +3  V               (d) +6  V

A.a

Given, I= 2 A.  Let VA and VB indicate potential differences at A and B.

Using Kirchoff’s voltage rule, we get,

Q. Current through the wire XY shown in the figure is

(a) 1 A                                    (b) 4 A                                   (c) 2 A                                    (d) A

A.c

Let i3 be the current flowing through the wire XY. Applying kirchhoff's rules upper left rule

Applying kirchhoff's rules upper right rule

Applying KVL in the outer loop containing 50 V battery,

Solving equations (i)  ,  (ii) , (iii)

## FAQs

Q. Are there any limitations of Kirchoff’s voltage law?
A. It is applicable only for constant currents in a circuit. When the current varies with time, a magnetic field and induced EMF is produced, and KVL becomes no longer valid.

Q. Can KVL be applied to an open circuit?
A. KVL deals with voltage increase or decrease and does not have anything to do with circuit being open or closed.

Q. Give one example where KVL is not applicable?
A. It is not applicable to high frequency applications like microwaves.

Q. Can Kirchoff’s rules be applied in nonlinear circuits?
A. Yes, Kirchoff’s rules can also be applied in nonlinear circuits like when diodes are present in a circuit.

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