Electric Circuits and Kirchhoff's Rules

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In many electrical circuits, Ohm's law cannot be applied. This happens when there is more than one source of emf in the circuit or when resistors are connected in a complicated manner. To solve such complex circuits, Gustav Robert Kirchhoff developed two laws based on charge neutrality in a metal.

Here, we first discuss the internal resistance of electrical circuits and then go on to Kirchhoff's rules.

Internal Resistance

When current is drawn from a cell, ions move within the cell from one electrode to another. The resistance offered by the electrodes and electrolytes to these, measure the internal resistance of the cell.The internal resistance of a cell depends on the distance between the plates, the nature of the electrolytes, the concentration of electrolytes, the nature of the electrodes and the area of the plates.

It is usual practice to represent internal resistance of a cell like a series resistor, external to the cell as shown.

Consider the circuit below.

Let e, r be the emf and internal resistance of a cell and R - the external resistance. A high resistance voltmeter V is connected as shown.

When K is opened (i.e., open circuit) emf, the voltmeter reads the emf (e) of the cell as no current flows through the circuit.

When K is closed (i.e., closed circuit), a current 'I' flows in the circuit. Hence, we have

'Ir' is the potential difference across the internal resistance r.

But, V = IR

Therefore, the external voltage V is less than the emf of the cell, e. It is as though an internal resistance r is in series with the external resistance R, and this determines the current in the circuit for a given source of emf.

Also,

Kirchhoff's Rules

Consider the following two circuits. Neither can be solved by series-parallel combinations.

Many practical resistor networks cannot be reduced to simple, series-parallel combinations. The above circuit shows a DC power supply with EMF E₁ charging a battery with a smaller EMF E₂ and feeding current to a light bulb with resistance R. Here we cannot identify the resistances in series or in parallel. So, German physicist Gustow Robert Kirchoff developed a technique. He introduced two terms. One is 'junction' and the other is 'loop'.

Before going on to Kirchoff's rules, we need to introduce two terms - junction and loop.

In the above circuits a, b, c, d are junctions but not e, f.

Some possible loops are acdba, acdefa, abdefa and abcdefa.

Kirchhoff's Junction Rule

The algebraic sum of the currents at a junction in a closed circuit is zero.

Therefore, I₁ + I₄ = I₂ + I₃ + I₅

Hence, I₁ + I₄ - I₂ - I₃ - I₅ = 0

or SI = 0

(Sum of currents entering a junction = Sum of currents leaving the junction)

This rule is based on the fact that charge cannot be accumulated at any point in a conductor in a steady situation.

Kirchhoff's Loop Rule

The algebraic sum of the potential differences in any loop including those associated with emfs and those of resistive elements must be equal to zero.

This rule is based on energy conservation, i.e., the net change in the energy of a charge after completing the closed path is zero. Otherwise, one can continuously gain energy by circulating charge in a particular direction.

Sign Convention in Applying Kirchhoff's Rules

The emf of a cell is positive when one moves in the direction of increasing potential (i.e., negative pole to positive pole) through the cell and is negative when one moves from positive to negative.

The product of resistance and current, i.e., the IR term, in any arm of the circuit is taken negative if one moves in a closed path, in the same direction of the assumed current; and positive if in the opposite direction.

Steps to solve circuits by Kirchhoff's laws:

• Assume unknown currents in a given circuit and show their directions by arrows.

• Choose any closed loop and find the algebraic sum of voltage drops plus the algebraic sum of the emfs in that closed loop and equate it to zero.

• Write equations for as many closed loops as the number of unknown quantities. Solve the equations to find the unknown quantities.

• If the value of assumed current is negative, it means that the actual direction of the current is opposite to that of the assumed direction.