conclusion for kirchhoff's law experiment

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To verify the Kirchhoff's voltage law and Kirchhoff's current law for the given circuit

Aim: verification of kirchoff's laws, apparatus required.

Circuit Diagrams

Given circuit.

1. KVL Circuit

Practical Circuit for KVL

2. KCL Circuit

Practical Circuit for KCL

Theory for Kirchhoff's Current and Kirchhoff's Volatage Law

a) Kirchhoff's Voltage law states that the algebraic sum of the voltage around any closed path in a given circuit is always zero. In any circuit, voltage drops across the resistors always have polarities opposite to the source polarity. When the current passes through the resistor, there is a loss in energy and therefore a voltage drop. In any element, the current flows from a higher potential to lower potential. Consider the fig shown above in which there are 3 resistors are in series. According to kickoff's voltage law.

V = V1 + V2 + V3

b) Kirchhoff's current law states that the sum of the currents entering a node equal to the sum of the currents leaving the same node. Consider the fig shown above in which there are 3 parallel paths. According to Kirchhoff's current law.

I = I1 + I2 + I3

Procedure for Kirchhoff's Voltage law:

1. Connect the circuit as shown in fig (2a).

2. Measure the voltages across the resistors.

3. Observe that the algebraic sum of voltages in a closed loop is zero.

Procedure for Kirchhoff's current law:

1. Connect the circuit as shown in fig (2b).

2. Measure the currents through the resistors.

3. Observe that the algebraic sum of the currents at a node is zero.

Observation Table for KVL

Observation Table for KCL

Precautions

• Avoid loose connections.
• Keep all the knobs in minimum position while switch on and off of the supply.

Viva Questions

• What is another name for KCL & KVL?
• Define network and circuit?
• What is the property of inductor and capacitor?

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21.3 Kirchhoff’s Rules

Learning objectives.

By the end of this section, you will be able to:

• Analyze a complex circuit using Kirchhoff’s rules, using the conventions for determining the correct signs of various terms.

Many complex circuits, such as the one in Figure 21.21 , cannot be analyzed with the series-parallel techniques developed in Resistors in Series and Parallel and Electromotive Force: Terminal Voltage . There are, however, two circuit analysis rules that can be used to analyze any circuit, simple or complex. These rules are special cases of the laws of conservation of charge and conservation of energy. The rules are known as Kirchhoff’s rules , after their inventor Gustav Kirchhoff (1824–1887).

Kirchhoff’s Rules

• Kirchhoff’s first rule—the junction rule. The sum of all currents entering a junction must equal the sum of all currents leaving the junction.
• Kirchhoff’s second rule—the loop rule. The algebraic sum of changes in potential around any closed circuit path (loop) must be zero.

Explanations of the two rules will now be given, followed by problem-solving hints for applying Kirchhoff’s rules, and a worked example that uses them.

Kirchhoff’s First Rule

Kirchhoff’s first rule (the junction rule ) is an application of the conservation of charge to a junction; it is illustrated in Figure 21.22 . Current is the flow of charge, and charge is conserved; thus, whatever charge flows into the junction must flow out. Kirchhoff’s first rule requires that I 1 = I 2 + I 3 I 1 = I 2 + I 3 (see figure). Equations like this can and will be used to analyze circuits and to solve circuit problems.

Making Connections: Conservation Laws

Kirchhoff’s rules for circuit analysis are applications of conservation laws to circuits. The first rule is the application of conservation of charge, while the second rule is the application of conservation of energy. Conservation laws, even used in a specific application, such as circuit analysis, are so basic as to form the foundation of that application.

Kirchhoff’s Second Rule

Kirchhoff’s second rule (the loop rule ) is an application of conservation of energy. The loop rule is stated in terms of potential, V V , rather than potential energy, but the two are related since PE elec = qV PE elec = qV . Recall that emf is the potential difference of a source when no current is flowing. In a closed loop, whatever energy is supplied by emf must be transferred into other forms by devices in the loop, since there are no other ways in which energy can be transferred into or out of the circuit. Figure 21.23 illustrates the changes in potential in a simple series circuit loop.

Kirchhoff’s second rule requires emf − Ir − IR 1 − IR 2 = 0 emf − Ir − IR 1 − IR 2 = 0 . Rearranged, this is emf = Ir + IR 1 + IR 2 emf = Ir + IR 1 + IR 2 , which means the emf equals the sum of the IR IR (voltage) drops in the loop.

Applying Kirchhoff’s Rules

By applying Kirchhoff’s rules, we generate equations that allow us to find the unknowns in circuits. The unknowns may be currents, emfs, or resistances. Each time a rule is applied, an equation is produced. If there are as many independent equations as unknowns, then the problem can be solved. There are two decisions you must make when applying Kirchhoff’s rules. These decisions determine the signs of various quantities in the equations you obtain from applying the rules.

• When applying Kirchhoff’s first rule, the junction rule, you must label the current in each branch and decide in what direction it is going. For example, in Figure 21.21 , Figure 21.22 , and Figure 21.23 , currents are labeled I 1 I 1 , I 2 I 2 , I 3 I 3 , and I I , and arrows indicate their directions. There is no risk here, for if you choose the wrong direction, the current will be of the correct magnitude but negative.
• When applying Kirchhoff’s second rule, the loop rule, you must identify a closed loop and decide in which direction to go around it, clockwise or counterclockwise. For example, in Figure 21.23 the loop was traversed in the same direction as the current (clockwise). Again, there is no risk; going around the circuit in the opposite direction reverses the sign of every term in the equation, which is like multiplying both sides of the equation by –1. –1.

Figure 21.24 and the following points will help you get the plus or minus signs right when applying the loop rule. Note that the resistors and emfs are traversed by going from a to b. In many circuits, it will be necessary to construct more than one loop. In traversing each loop, one needs to be consistent for the sign of the change in potential. (See Example 21.5 .)

• When a resistor is traversed in the same direction as the current, the change in potential is − IR − IR . (See Figure 21.24 .)
• When a resistor is traversed in the direction opposite to the current, the change in potential is + IR + IR . (See Figure 21.24 .)
• When an emf is traversed from – – to + (the same direction it moves positive charge), the change in potential is +emf. (See Figure 21.24 .)
• When an emf is traversed from + to – – (opposite to the direction it moves positive charge), the change in potential is − − emf. (See Figure 21.24 .)

Example 21.5

Calculating current: using kirchhoff’s rules.

Find the currents flowing in the circuit in Figure 21.25 .

This circuit is sufficiently complex that the currents cannot be found using Ohm’s law and the series-parallel techniques—it is necessary to use Kirchhoff’s rules. Currents have been labeled I 1 I 1 , I 2 I 2 , and I 3 I 3 in the figure and assumptions have been made about their directions. Locations on the diagram have been labeled with letters a through h. In the solution we will apply the junction and loop rules, seeking three independent equations to allow us to solve for the three unknown currents.

We begin by applying Kirchhoff’s first or junction rule at point a. This gives

since I 1 I 1 flows into the junction, while I 2 I 2 and I 3 I 3 flow out. Applying the junction rule at e produces exactly the same equation, so that no new information is obtained. This is a single equation with three unknowns—three independent equations are needed, and so the loop rule must be applied.

Now we consider the loop abcdea. Going from a to b, we traverse R 2 R 2 in the same (assumed) direction of the current I 2 I 2 , and so the change in potential is − I 2 R 2 − I 2 R 2 . Then going from b to c, we go from – – to +, so that the change in potential is + emf 1 + emf 1 . Traversing the internal resistance r 1 r 1 from c to d gives − I 2 r 1 − I 2 r 1 . Completing the loop by going from d to a again traverses a resistor in the same direction as its current, giving a change in potential of − I 1 R 1 − I 1 R 1 .

The loop rule states that the changes in potential sum to zero. Thus,

Substituting values from the circuit diagram for the resistances and emf, and canceling the ampere unit gives

Now applying the loop rule to aefgha (we could have chosen abcdefgha as well) similarly gives

Note that the signs are reversed compared with the other loop, because elements are traversed in the opposite direction. With values entered, this becomes

These three equations are sufficient to solve for the three unknown currents. First, solve the second equation for I 2 I 2 :

Now solve the third equation for I 3 I 3 :

Substituting these two new equations into the first one allows us to find a value for I 1 I 1 :

Combining terms gives

Substituting this value for I 1 I 1 back into the fourth equation gives

The minus sign means I 2 I 2 flows in the direction opposite to that assumed in Figure 21.25 .

Finally, substituting the value for I 1 I 1 into the fifth equation gives

Just as a check, we note that indeed I 1 = I 2 + I 3 I 1 = I 2 + I 3 . The results could also have been checked by entering all of the values into the equation for the abcdefgha loop.

Problem-Solving Strategies for Kirchhoff’s Rules

• Make certain there is a clear circuit diagram on which you can label all known and unknown resistances, emfs, and currents. If a current is unknown, you must assign it a direction. This is necessary for determining the signs of potential changes. If you assign the direction incorrectly, the current will be found to have a negative value—no harm done.
• Apply the junction rule to any junction in the circuit. Each time the junction rule is applied, you should get an equation with a current that does not appear in a previous application—if not, then the equation is redundant.
• Apply the loop rule to as many loops as needed to solve for the unknowns in the problem. (There must be as many independent equations as unknowns.) To apply the loop rule, you must choose a direction to go around the loop. Then carefully and consistently determine the signs of the potential changes for each element using the four bulleted points discussed above in conjunction with Figure 21.24 .
• Solve the simultaneous equations for the unknowns. This may involve many algebraic steps, requiring careful checking and rechecking.
• Check to see whether the answers are reasonable and consistent. The numbers should be of the correct order of magnitude, neither exceedingly large nor vanishingly small. The signs should be reasonable—for example, no resistance should be negative. Check to see that the values obtained satisfy the various equations obtained from applying the rules. The currents should satisfy the junction rule, for example.

The material in this section is correct in theory. We should be able to verify it by making measurements of current and voltage. In fact, some of the devices used to make such measurements are straightforward applications of the principles covered so far and are explored in the next modules. As we shall see, a very basic, even profound, fact results—making a measurement alters the quantity being measured.

Check Your Understanding

Can Kirchhoff’s rules be applied to simple series and parallel circuits or are they restricted for use in more complicated circuits that are not combinations of series and parallel?

Kirchhoff's rules can be applied to any circuit since they are applications to circuits of two conservation laws. Conservation laws are the most broadly applicable principles in physics. It is usually mathematically simpler to use the rules for series and parallel in simpler circuits so we emphasize Kirchhoff’s rules for use in more complicated situations. But the rules for series and parallel can be derived from Kirchhoff’s rules. Moreover, Kirchhoff’s rules can be expanded to devices other than resistors and emfs, such as capacitors, and are one of the basic analysis devices in circuit analysis.

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Kirchhoff’s Law

Introduction: what is kirchhoff’s law.

Kirchhoff’s laws are a set of laws that quantify how current flows through a circuit and how voltage varies around a loop in a circuit. They are used to govern the conservation of charge and energy in standard electrical circuits. Two significant circuital laws are applied in every simple and complex electrical circuit in physics. These laws were postulated in 1845 by German physicist Gustav Kirchhoff. The proof of Kirchhoff’s law can be obtained by using Maxwell’s equations.

Kirchhoff’s Current Law (KCL) or Kirchhoff’s 1 st Law

Current is defined as the rate of change of charge passing through a conducting wire. Kirchhoff’s current law states that the “total current entering a junction or node is exactly equal to the current leaving the node.” In other words, the algebraic sum of all the currents entering the node must be equal to the algebraic sum of all the currents leaving a node. This law is commonly known as the conservation of charge. The formula is given by

Σ I in = Σ I out

Kirchhoff’s Voltage Law (KVL) or Kirchhoff’s 2 nd Law

Electric potential represents the concentration of energy in a circuit. The potential quickly spreads to a uniform value throughout an uninterrupted section of the electrical circuit. The difference in electric potential is called the voltage.

Kirchhoff’s voltage law states that “in any closed-loop network, the sum of voltage drops around the loop is equal to zero.” This law is known as the conservation of energy . The formula is given by

Σ V total = 0

The term node in an electrical circuit generally refers to a connection or junction of two or more current-carrying paths. Also, for current to flow either in or out of a node, a closed-circuit path must exist.

Differences Between Kirchhoff’s Current and Voltage Laws

Kirchhoff’s Law Circuit Diagram

A circuit diagram consists of a source of current and voltage along with resistances and impedances, which can be in series, or parallel, or combination of the two. The polarity of the source is indicated by positive and negative signs, which automatically applies to the resistances.

Resistances in Series

Resistances are said to be in series when they are connected in a single path. The current from a source flows through all the resistances in a closed loop.

Resistances in Parallel

Resistances are said to be in parallel when the path branches and each branch consists of one resistance. The current from the source splits into different paths. The equation for replacing resistances in parallel is a bit more complicated.

Sign Convention

The sign convention for applying signs to the voltage polarities in KVL equations is as follows. When traversing the loop, if the positive terminal of a voltage difference is encountered before the negative terminal, the voltage difference will be interpreted as positive. If the negative terminal is encountered first, the voltage difference will be interpreted as negative.

Applications of Kirchhoff’s Law

Kirchhoff’s laws are applicable to analyze any circuit regardless of the composition and structure of it. Some of its applications include

• To find the unknown resistances, impedances, voltages, and currents (direction as well as value).
• In a branched circuit, currents passing each branch are determined by applying KCL at every junction and KVL in every loop.
• In a looped circuit, the current passing each independent loop is determined by applying KVL for each loop and calculating the currents in any resistance of the circuit.

Limitations of Kirchhoff’s Law

Kirchhoff’s laws are limited in their applicability. They are valid for all cases in which total electric charge is constant in the region into consideration. Essentially, this is always true, so long as the law is applied for a specific point. Over a region, however, charge density may not be constant. Because the charge is conserved, the only way this is possible is if there is a flow of charge across the boundary of the region. This flow would result in current, thus violating Kirchhoff’s laws.

Another limitation is that it works under the assumption that there is no fluctuating magnetic field in the closed-loop. Electric fields and emf could be induced, which causes Kirchhoff’s laws to break in the presence of a variable magnetic field.

• Kirchhoff’s Voltage Law (KVL) – Allaboutcircuits.com
• Kirchhoff’s Current Law and Kirchhoff’s Voltage Law – Circuitglobe.com
• Kirchhoff’s Voltage Law and Kirchhoff’s Current Law – Ultimateelectronicsbook.com
• Kirchhoff’s Current Law and Kirchhoff’s Voltage Law – Electrical4u.com
• Kirchhoff’s Law  – Web.pa.msu.edu
• Kirchhoff’s Circuit Laws –   Web.iit.edu
• Kirchhoff’s Law –   Vlab.amrita.edu
• Kirchhoff’s Laws – Theory.uwinnipeg.ca
• Kirchhoff’s Laws – Isaacphysics.org
• Kirchhoff’s Laws – Cpp.edu
• Kirchhoff’s Circuit Laws – Electronics-tutorials.ws
• Kirchhoff’s Laws: KCL & KVL – Electronicshub.org
• Kirchhoff’s Law – Resistorguide.com
• How Kirchhoff’s Law Works – Elprocus.com

Article was last reviewed on Friday, February 3, 2023

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