Maharashtra Board Class 12 Physics Chapter 9 Current Electricity PDF Download

9. Current Electricity

Can You Recall?

There can be three types of electrical conductors: good conductors (metals), semiconductors and bad conductors (insulators).

Does a semiconductor diode and resistor have similar electrical properties?

Can you explain why two or more resistors connected in series and parallel have different effective resistances?

9.1 Introduction

In XIth Std. we have studied the origin of electrical conductivity, in particular for metals. We have also studied how to calculate the effective resistance of two or more resistances in series and in parallel. However, a circuit containing several complex connections of electrical components cannot be easily reduced into a single loop by using the rules of series and parallel combination of resistors.

More complex circuits can be analyzed by using Kirchhoff's laws. Gustav Robert Kirchhoff (1824-1887) formulated two rules for analyzing a complicated circuit. In this chapter we will discuss these laws and their applications.

9.2 Kirchhoff's Laws Of Electrical Network

Before describing these laws we will define some terms used for electrical circuits.

Junction: Any point in an electric circuit where two or more conductors are joined together is a junction.

Loop: Any closed conducting path in an electric network is called a loop or mesh.

Branch: A branch is any part of the network that lies between two junctions.

In Fig. 9.1, there are two junctions, labeled a and b. There are three branches: these are the three possible paths 1, 2 and 3 from a to b.

For a steady current flowing through an electrical network of resistors, the following Kirchhoff's laws are applicable.

9.2.1 Kirchhoff's First Law: (Current Law / Junction Law)

The algebraic sum of the currents at a junction is zero in an electrical network, i.e., \(\sum_{i=1}^{n} I_i = 0\), where \(I_i\) is the current in the ith conductor at a junction having n conductors.

Sign Convention: The currents arriving at the junction are considered positive and the currents leaving the junction are considered negative.

Consider a junction P in a circuit where six conductors meet (Fig. 9.2). Applying the sign convention, we can write:

\(I_1 - I_2 + I_3 + I_4 - I_5 - I_6 = 0\) --- (9.1)

Arriving currents \(I_1\), \(I_3\) and \(I_4\) are considered positive and leaving currents \(I_2\), \(I_5\) and \(I_6\) are considered negative.

Equation (9.1) can also be written as \(I_1 + I_3 + I_4 = I_2 + I_5 + I_6\)

Thus the total current flowing towards the junction is equal to the total current flowing away from the junction.

Teacher's Note

Kirchhoff's first law helps you understand how electricity flows in circuits. Think of it like water flowing in pipes - water coming in must equal water going out at any point.

Exam Trick

Remember: Currents coming in = Currents going out. Just like money in your bank account - money deposited must equal money withdrawn.

Points To Remember

Junction is a point where two or more wires meet in a circuit.
At any junction, total incoming current equals total outgoing current.
Kirchhoff's first law is based on conservation of electric charge.

9.2.2 Kirchhoff's Voltage Law

The algebraic sum of the potential differences (products of current and resistance) and the electromotive forces (emfs) in a closed loop is zero.

\(\sum IR + \sum \varepsilon = 0\) --- (9.2)

Sign Convention:

1. While tracing a loop through a resistor, if we are travelling along the direction of conventional current, the potential difference across that resistance is considered negative. If the loop is traced against the direction of the conventional current, the potential difference across that resistor is considered positive.

2. The emf of an electrical source is positive while tracing the loop within the source from the negative terminal of the source to its positive terminal. It is taken as negative while tracing within the source from positive terminal to the negative terminal.

Consider an electrical network shown in Fig. 9.3.

Consider the loop ABFGA in clockwise sense. Applying the sign conventions to Eq. (9.2), we get:

\(-I_1 R_1 - I_3 R_5 - I_1 R_3 + H_1 = 0\)

\(\therefore H_1 = I_1 R_1 + I_3 R_5 + I_1 R_3\)

Now consider the loop BFDCB in anticlockwise direction. Applying the sign conventions, we get:

\(-I_2 R_2 - I_3 R_5 - I_2 R_4 + H_2 = 0\)

\(\therefore H_2 = I_2 R_2 + I_3 R_5 + I_2 R_4\)

Teacher's Note

Kirchhoff's voltage law tells us that energy is conserved in a circuit. All the voltage supplied by batteries must be used up by resistors.

Exam Trick

Remember: Sum of voltages around any closed loop = zero. Like going around a circle and coming back to the same point.

Points To Remember

Voltage law applies to any closed loop in a circuit.
The sum of all voltage increases and decreases in a loop is zero.
This law is based on conservation of energy.
Direction matters when using sign convention.

Steps Usually Followed While Solving A Problem Using Kirchhoff's Laws

i) Choose some direction of the currents.

ii) Reduce the number of variables using Kirchhoff's first law.

iii) Determine the number of independent loops.

iv) Apply voltage law to all the independent loops.

v) Solve the equations obtained simultaneously.

vi) In case, the answer of a current variable is negative, the conventional current is flowing in the direction opposite to that chosen by us.

Example 9.1: Figure Shows Currents In A Part Of Electrical Circuit. Find The Current X?

Solutions: At junction B, current \(I_1\) is split into \(I_2\) and \(I_3\) therefore \(I_1 = I_2 + I_3\)

Substituting values we get \(I_3 = 14\) A

At C, \(I_5 = I_3 + I_4\) therefore \(I_5 = 16\) A

At D, \(I_5 = I_6 + I_7\) therefore \(I_6 = 7\) A

Kirchhoff's first law is consistent with the conservation of electrical charge while the voltage law is consistent with the law of conservation of energy.

Some charge is received per unit time due to the currents arriving at a junction. For conservation of charge, same amount of charge must leave the junction per unit time which leads to the law of currents.

Algebraic sum of emfs (energy per unit charge) corresponds to the electrical energy supplied by the source. According to the law of conservation of energy, this energy must appear in the form of electrical potential difference across the electrical elements/devices in the loop. This leads to the law of voltages.

Teacher's Note

These examples show real problems students can solve using Kirchhoff's laws. Practice finding currents at different junctions step by step.

Exam Trick

Always start at a simple junction with only a few currents. Write one equation at a time and solve step by step.

Points To Remember

First find the junction with fewest unknown currents.
Apply Kirchhoff's first law at that junction.
Move to the next simple junction and repeat.
Check your answer - all currents must balance.

9.3 Wheatstone Bridge

Resistance of a material changes due to several factors such as temperature, strain, humidity, displacement, liquid level, etc. Therefore, measurement of these properties is possible by measuring the resistance. Measurable values of resistance vary from a few milliohms to hundreds of mega ohms.

Depending upon the resistance range (milliohm to tens of ohm, tens of ohm to hundreds of ohms, hundreds of ohm to mega ohm, etc.), various methods are used for resistance measurement. Wheatstone's bridge is generally used to measure resistances in the range from tens of ohm to hundreds of ohms.

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