Question

derivation of resistor in series​

Answer 1

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• derivation of resistor in series

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$\small\mathfrak\fcolorbox{lime}{black}{\pink{\underline{\red{\underline\blue{Series combination of resistors}}}}}$

when two or more resistors are combined together end to end. This combination is called series combination of resistors.

In a figure shows the series combination of three resistors of resistance R1, R2 and R3 respectively. Equal current I ampere is flowing through each resistor but potential difference produce across different registers are different as V1, V2 and V3.

we know that in series combination: Applied potential = sum of all P.D across

different registers

V = V1 + V2 + V3 --------(1)

By ohm's law:

P.D across first resistor, V1 = R1×I

P.D across second resistor, V2 = R2×I

P.D across third resistor, V3 = R3×I

If the equivalent resistance of the combination be “R” ohm. Thus we can replace all the resistance by single resistor of resistance R.

then,

potential difference across equivalent register, V = R×I

All these value put in eqn1

R×I = R1×I + R2×I + R3×I

R×I = I ( R1+ R2+ R3)

[ R = R1+ R2+ R3 ]

$\small\mathbb\fcolorbox{lime}{black}{\pink{\underline{\red{\underline\blue{Parallel combination of resistors !}}}}}$

when two or more resistors are joined together between to fixed point with battery or cell. Thus it is called parallel combination of resistors.

In above figure shows the parallel combination of three resistors of resistance R1, R2 and R3 respectively. In this combination the main current I divide in three part I1, I2 and I3 at the junction point that is different amount of current flowing through the registers but the potential difference across them be same as V volt.

It is clear from the circuit:

Total current at junction point:

I = I1 + I2 + I3 -------(1)

By ohm's law

V = RI

Hence, I = V/R

Thus,

current in 1st resistor, I1 = V/R1

current in 2nd resistor, I2 = V/R2

current in 3rd resistor, I3 = V/R3

If the equivalent resistance of this combination be “R”omh, then current flowing in equivalent resistance:

I = V/R

All these values put in eqn1

V/R = V/R1 + V/R2 + V/R3

V/R = V (1/R1 + 1/R2 + 1/R3)

[ 1/R = 1/R1 + 1/R2 + 1/R3 ]

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