Resistors and Resistivity

1.0Resistors

A physical device which has the principal characteristic of offering electric resistance is called ‘resistor’.

Materials used for resistors

(1) Alloys like manganin, constantan, nichrome, etc.; used in wire wound resistors. 

(2) Carbon resistors ; compact and low cost.

(3) Aluminium or copper wires; low resistance conductors used to make connecting wires/electrical transmission lines. 

The resistors which obey Ohm’s law are called linear resistors. The resistors which do not obey Ohm’s law are called non-linear resistors.

Resistance ‘R’ is not a material property, that is, its value changes from sample to sample for a given material. Resistance depends on the nature of the substance, its shape and size (geometrical factors like length, cross-sectional area).

Resistance is directly proportional to the length (l) of a conductor and inversely proportional to its area of cross-section (A).$R\propto l$

$R\propto \frac{l}{A}$

Where, ρ is a constant called resistivity of material.

Resistance ‘R’ depends on :  

(1) Length of conductor

(2) Cross-sectional area  

(3) Type of material

(4) Temperature

Wider wires have a greater cross-sectional area. Water will flow through a wider pipe at a higher rate than it will flow through a narrow pipe. This can be attributed to the lower amount resistance that is present in the wider pipe and vice-versa. In the same manner, the wider the wire, the less resistance there will be to the flow of the electric charge.

2.0Resistivity (Specific Resistance)

It is a characteristic property of a material rather than that of a particular specimen of a material. It depends on physical conditions such as temperature and pressure. 

If l is 1 unit length and A is 1 square unit area, then, R = ρ

Resistivity of a conductor is ‘the resistance offered by a uniform conducting wire having unit length and unit area of cross-section’.

Unit of Resistivity

SI unit : ohm-meter (Ωm)

Resistance and resistivity of substances depend on temperature. For metals, they increase with the increase in temperature. For insulators or semiconductors, they decrease with the increase in temperature. 

For a wire of cylindrical cross-section,

$Radius=\frac{Diameter}{2}$

Area of cross section of a circular wire = πr²

1. Let the resistance of an electrical component remain constant while the potential difference across the two ends of the component decreases to half its former value.  What change will occur in the current through it?

Solution

We know that $I=\frac{V}{R}$, when potential difference becomes$\frac{V}{2}$, and resistance remains constant, then, current becomes $\frac{1}{2}$ of its former value.  

2. When a 12 V battery is connected across an unknown resistor, there is a current of 2.5 mA in the circuit. Find the value of the resistance of the resistor.

Solution

Here, V = 12 V, I = 2.5 mA = 2.5 × 10^–3 A

Resistance of the resistor, 

$R=\frac{V}{I}=\frac{12}{2.5\times 10^{-3}}=4800\; \Omega =4.8k\Omega$

3.0Effect of stretching of a wire on resistance

Let a wire of length $l_1$ and cross-sectional area A₁ be stretched to a length $l_2$ and its cross-sectional area becomes A₂ such that$l_2=n{l}_1$ (see figure). 

Now, volume after stretching = volume before stretching

or $l_2{A}_2=l_1{A}_{1}or(n{l}_1)A_2=l_1{A}_1$

or  $A_2=\frac{A_1}{n}$ .... (1)

Initial resistance, $R_1=\rho \frac{l_1}{A_1}$ .... (2)

Final resistance, $R_2=\rho \frac{l_2}{A_2}=\rho \frac{n{l}_1}{(A_1/n)}=n^2\left(\rho \frac{l_1}{A_1}\right)$ [Using (1) and (2)]

or $R_2=n^2{R}_1$

1. An 24 Ω resistance wire is doubled on itself. Calculate the value of the new resistance offered by the wire.

Let the length and area of cross-section of the original wire be l and A respectively
(see figure). 

Original resistance,

$R=\rho \frac{l}{A}$ …(1)

When a wire is doubled on itself, its length becomes halved and area of cross-section becomes doubled i.e., the new length, $l^{\prime }=\frac{l}{2}$ and a new area of cross-section, A′ = A + A = 2 A.       

New resistance,$R^{\prime }=\rho \frac{l^{\prime }}{A^{\prime }}=\rho \frac{(l/2)}{2A}=\frac{1}{4}\left(\rho \frac{l}{A}\right)=\frac{1}{4}R=\frac{1}{4}\times 24=6\Omega$

2. Two materials have different resistivities. Two wires of the same length are made, one from each of the materials. Is it possible for each wire to have the same resistance?

Two wires of the same length are made, one from each of the materials. The resistance of a wire is given by, $R=\rho \frac{l}{A}$ , where ρ is the resistivity of the wire material, and  and A are respectively, the length and cross-sectional area of the wire. Even when the wires have the same length, they may have the same resistance, if the cross-sectional areas of the wires are chosen so that the ratio $\rho \frac{l}{A}$ is the same for each. 

That is, ${\rho }_1\frac{l}{A_1}={\rho }_2\frac{l}{A_2}or\frac{A_2}{A_1}=\frac{{\rho }_2}{{\rho }_1}$

This is the condition for each wire of different materials to have the same resistance when they have same length. 

Two wires made of iron of different sizes may have different resistances, but they have same resistivities. This is because resistance differ from specimen to specimen while resistivity is constant for a given material at a given temperature.

Also Read