Resistance

A physicist Georg Simon Ohm found out the relation between the flowing current I, which is flowing in a metallic wire having the potential difference across its terminals (ends). He stated that: The potential difference in volts V, across the terminals of a given metallic wire in an electric circuit is directly proportional to the current flowing I through it, given its temperature remains the same.

This is known as Ohm’s law. In other words, it can be written as: Potential difference across the ends ∝ Current

or 

V ∝ I

V = I R, where R is constant of proportionality called Resistance.

Resistance is a property that opposes (resists) the flow of current, or it can be said that the flow of electrons in a conductor. It controls the magnitude (value) of the current flowing through the circuit.

The SI unit of resistance is measured in ohm, which is denoted by Ω.

Factors on which Resistance depends

Resistance of a conductor depends on:

• Its length (l),
• Its area of cross-section (A) and
• The nature of its material.

The resistance of the uniformly distributed metallic conductor is directly proportional to its length and inversely proportional to the area of its cross-section. 

Therefore, R ∝ l and R ∝ 1/A.

⇒ R ∝ l / A

or 

⇒ R = ρl / A                           

where ρ is the constant of proportionality known as electrical resistivity of the conductor material. 

• The SI unit is Ω.m. 
• Resistivity varies with temperature (T).
• The value of resistivity of conductors is very low whereas the insulators have very high resistivity. 
• Alloys having higher resistivity than metals are used in electrical heating devices, like iron and toasters, copper and aluminum are used for electrical transmission lines, and tungsten is used in filament of electric bulbs.
• Hence, Resistivity of Insulators > Resistivity of Alloys > Resistivity of Conductors.

Combination of Resistors

Resistors are used in various combinations. There are two methods of arranging the resistors in different combinations:

(i) Resistors in Series

(ii) Resistors in Parallel

Resistors in Series Combination: 

Two or more resistances are said to be connected in series when they are connected end to end and the same current flows through each of them in turn. In this case, the equivalent or the total resistance equals the sum of the number of individual resistances present in the series combination.
Mathematically, the equivalent resistance of any number of resistances (R₁, R₂, R₃, R₄, R₅, ……..) connected in series is given as:
R_eq = R₁ + R₂ + R₃ + R₄ + R₅ + ……..

The equivalent current flow through it is I, detected through the ammeter A and key K.

The equivalent potential difference is equal to the sum of the individual potential difference across each resistor, i.e.

V_eq = V₁ + V₂ + V₃

The current I through each resistor is the same i.e. I = I₁ = I₂ = I₃

Replace the three resistors connected in series by an equivalent single resistor of resistance R_eq, such that the potential difference V_eq across its terminals, and the current I through the circuit remains the same. 

Applying Ohm’s law to the circuit:

V_eq = IR_eq

By applying Ohm’s law to all resistors individually as:

V₁ = IR₁

V₂ = IR₂

V₃ = IR₃

Hence, IR= IR₁ + IR₂ + IR₃

or

R_eq = R₁ + R₂ + R₃

Resistors in Parallel Combination:

Two or more resistances are said to be connected in parallel connected when they are connected between two points and each has a different current direction. The current is branched out and recombined as the branches intersect at a common point in such circuits. 

Mathematically, the equivalent resistance of any number of resistances (R₁, R₂, R₃, R₄, R₅, ……..) connected in parallel is given as:

1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + 1/R₄ + 1/R₅ + ……..

Consider a case of three resistances (R₁, R₂, and R₃) connected in parallel with each other with the corresponding voltage source (V₁, V₂, and V₃) in a circuit shown below:

Here, the current flows through each resistor is different therefore, the equivalent current flown through the circuit is:

I_eq = I₁ + I₂ + I₃

Replace the three resistors connected in parallel by an equivalent single resistor of the parallel combination of resistors be R_eq.

Now, by applying Ohm’s law to the parallel combination of resistors as:

I_eq = V / R_eq

On applying Ohm’s law to individual resistors as:

I₁ = V / R₁ 

I₂ = V / R₂

I₃ = V / R₃ 

Hence, V / R_eq = V / R₁ + V / R₂ + V / R₃ 

or   

1 / R_eq = 1 / R₁ + 1 / R₂ + 1 / R₃