The parallel circuit is one in which several resistance are connected across one another in such away that one terminal of each is connected to form a junction point while the remaining ends are also joined to form another junction

Consider a parallel circuit shown in the Fig. 1

In the parallel connection shown, the three resistances R

_{1}, R_{2 }and R_{3 }are connected in parallel and combination is connected across a source of voltage 'V'. In parallel circuit current passing through each resistance is different. Let total current drawn is say ' I ' as shown. There are three paths for this current, one through R

Now let us study current distribution. Apply Ohm's law to each resistance._{1 }, second through R_{2 }and third through R_{3 }. Depending up on the values of R_{1}, R_{2 }and R_{3 }the appropriate fraction of total current passes through them. These individual currents are shown as I_{1,}I_{2 }and I_{3}.While the voltage across the tow ends of each resistances R_{1 }, R_{2 }and R_{3 }is the same and equals the supply voltage V.Fig. 1 |

V= I

_{1 }R

_{1 }, V= I

_{2 }R

_{2 }, V = I

_{3 }R

_{3}

I

_{1 }= V/R

_{1 }, I

_{2 }= V/R

_{2 }I

_{3 }= V/R

_{3}

I = I

_{1 }+ I

_{2 }+I

_{3 }= V/R

_{1 }+ V/R

_{2 }+ V/R

_{3 }

= V{ 1/R

_{1 }+ 1/R

_{2 }+ 1/R

_{3 }) .................... (1)

For overall circuit if Ohm's law is applied,

V = I R

_{eq }

I = V/R

_{eq }......................(2) Where R

_{eq }= Total or equivalent resistance of the circuit. Comparing the tow equations,

Where R is the equivalent resistance of the parallel combination.

In general if 'n' resistance are connected in parallel,

__1.1 Conductance (G) :__

It is known that, 1/R = G (Conductance) hence,

**.**G = G

^{.}._{1 }+G

_{2 }+G

_{3 }+.............+ G

_{n }......For parallel circuits

**Important result :**

Now if n = 2, tow resistance are in parallel then,

The formula is directly used hereafter, for tow resistance in parallel.

__1.2 Characteristic of Parallel Circuits__

2) The total current gets divided into the number of paths equal to the number of resistances in parallel. The total current is always sum of all the individual currents.

I = I

_{1 }+ I

_{2 }+I

_{3 }+ ..........+ I

_{n }

3) The reciprocal of the equivalent resistance of a parallel circuit is equal to the sum of the reciprocal of the individual resistances.

4) The equivalent resistances is the smallest of all the resistances.

R< R

_{1 }, R < R

_{2 },.............., R < R

_{n }

5) The equivalent conductance is the arithmetic addition of the individual conductances.

**Key point**: The equivalent resistance is smaller than the smallest of all the resistances connected in parallel.

__2. Inductors in Parallel__ Consider the Fig. 1(a). Tow inductors L

_{1 }and L_{2}are connected in parallel. The current flowing through L_{1 }and L_{2}are i_{1 }and i_{2 }respectively. The voltage developed across L_{1 }and L_{2}are V_{L1 }and V_{L2 }respectively. the equivalent circuits shown in Fig. 2(b).Fig. 2 |

For inductor we have,

For parallel combination,

V = V

_{L1 }= V

_{L2 }= and

i = i

_{1}+ i

_{2}

That means, reciprocal of equivalent inductance of the parallel combination is the sum of reciprocals of the individual inductances

For n inductances in parallel,

__3. Capacitors in Parallel__ Consider the Fig. 3(a). Tow capacitors C

The equivalent circuit is shown in the Fig. 3(b)._{1 }and C_{2 }are connected in parallel. The current flowing through C_{1 }and C_{2 }are i_{1}and i_{2}respectively and voltages developed across are V_{C1 }and V_{C2 }respectively.Fig. 3 |

For capacitor we have,

For parallel combination,

V

_{C1 }= V

_{C2 }= V

_{C }and

i = i

_{1}+ i

_{2}

_{ }

C

_{eq }= C

_{1 }+ C

_{2 }

That means, equivalent capacitance of the parallel combination of the capacitance is the sum of the individual capacitance connected in series.

For n capacitor in parallel,**.**

^{.}.**C**

_{eq }= C_{1 }+ C_{2 }+ ........+ C_{n}
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