Difference Between Resistance And Resistivity

Resistance and resistivity provide essential insights into the behavior of electrical circuits, influencing everything from the design of electronic devices to the selection of materials for electrical applications. Resistance refers to the opposition a material offers to the flow of electric current. Resistivity, on the other hand, is an intrinsic property of a material that quantifies its ability to resist current flow, regardless of its shape or size.

1.0Resistance

• The resistance of a conductor refers to its opposition to the flow of electric charge. When a potential difference is applied across the conductor, free electrons are accelerated and collide with positive ions, which hinders their motion. This opposition provided by the ions is what we term the resistance of the conductor.
• Unit : ohm, volt/ampere    
• Dimensions = [M L² T^–3 A^–2]

Cause Of Resistance

• When a conductor is connected across a cell or battery ,free electrons in the conductors flows from negative to positive endpoint of the battery. This give rise to the electric current in the conductor. When free electrons flows from one end to another end of the conductor they collide with the ions of the conductors. This collision offer opposition or resistance to the flow of electrons through the conductors.

Factors On Which Resistance Depends

1.  Length of the conductor $R\propto l$
2.  Area of cross-section of the conductor $R\propto \frac{1}{A}$
3. Nature of material of the conductor $R=\rho \frac{l}{A}$
4. Temperature
5. Resistance of the conductor also depends on the direction of flow of current, $R=\rho \frac{l}{A}$

Where ℓ is the length of conductor along which current is flowing And A is area of cross-sectional of conductor perpendicular to the direction of current flow

Key points:

1. On stretching l increases, A decreases, R increases
2. If percentage change in length is less than 5% than percentage fractional change in resistance of wire will be

$\frac{\Delta R}{R}\times 100\%=2\left(\frac{\Delta l}{l}\right)\times 100\%\text{ (use }\frac{\Delta l}{l}\text{ with sign) }$

3. Similarly for Area

$\frac{\Delta R}{R}\times 100\%=-2\left(\frac{\Delta A}{A}\right)\times 100\%(\text{ use }\frac{\Delta A}{A}\text{ with sign) }$

4. Similarly for Radius

$\frac{\Delta R}{R}\times 100\%=-4\left(\frac{\Delta r}{r}\right)\times 100\%\text{ (use }\frac{\Delta r}{r}\text{ with sign) }$

5. If percentage increase in dimensions is greater than 5% than percentage fractional change in resistance will be 

$\frac{\Delta R}{R}\times 100\%=\frac{R_f-R_i}{R_i}\times 100\%$

Effect Of Stretching On Resistance Of Wire

Before Stretching $V_1=A_1{l}_1$

After Stretching $V_2=A_2{l}_2$

Volume of wire remains the same in stretching.

$A_1{l}_1=A_2{l}_2$

Initial Resistance of wire : $R_1=\frac{\rho l_1}{A_1}$

Final Resistance of wire : $R_2=\frac{\rho l_2}{A_2}$

$\frac{R_1}{R_2}=\frac{\rho l_1}{A_1}\times \frac{A_2}{\rho l_2}=\frac{l_1}{l_2}\times \frac{A_2}{A_1}=\frac{l_1^2}{l_2^2}$

$R\propto l^2$

If $l_2=n{l}_1$

$\frac{R_2}{R_1}={\left(\frac{l_2}{l_1}\right)}^2={\left(\frac{n{l}_1}{I_1}\right)}^2=n^2{R}_2=n^2{R}_1$

Also

$\frac{R_2}{R_1}={\left(\frac{l_2}{l_1}\right)}^2={\left(\frac{A_1}{A_2}\right)}^2={\left(\frac{r_1}{r_2}\right)}^4$

$R\propto \frac{1}{A^2}\propto \frac{1}{r^4}\propto \frac{1}{D^4}$

2.0Resistivity or Specific Resistance

The resistivity (specific resistance) of a material is equal to the resistance of a wire of that material with unit cross sectional area and unit length.

$\rho =\frac{RA}{l}\text{ if }l=1\text{ units then }\rho =R\text{ Units }$

From Ohm’s Law

$R=\frac{V}{I}=\frac{El}{neA\left(\frac{Ee\tau }{m_e}\right)}=\frac{m_{e}l}{n{e}^{2}A\tau }$

$R=\left(\frac{m}{n{e}^2\tau }\right)\times \frac{l}{A}=\frac{\rho l}{A}$

$\rho =\left(\frac{m}{n{e}^2\tau }\right)\text{,Resistivity }$

• S I Unit of Resistivity is=m
• Conductivity is the inverse of Resistivity $\sigma =\frac{1}{\rho }$
• Conductivity is the extent to which a material conducts electricity.

3.0Factors Affecting Resistivity ($\rho$) and Conductivity ($\sigma$)

1. Nature Of Material
2. Temperature Of Material
3. Resistivity and conductivity does not depend on the size and shape of the material because it is a characteristic property of the conducting material.

4.0Solved Examples 

Q-1. Find $R_{12},R_{34},R_{56}$ on the basis of the direction of current flow through different paths.      

Solution:

$R_{12}=\frac{\rho (3a)}{2a^2}=\frac{3\rho }{2a}$

$R_{34}=\frac{\rho (a)}{6a^2}=\frac{\rho }{6a}$

$R_{56}=\frac{\rho (2a)}{3a^2}=\frac{2\rho }{3a}$

$R_{12}>R_{56}>R_{34}$

Q-2. Find out the resistance of the current flowing through hollow cylindrical conductor

Solution: Electric current is flowing along the length of conductor through area of cross section 

$A=\pi \left(r_2^2-r_1^2\right)$

If l is the length of conductor along current flow 

$R=\rho \frac{l}{A}\Rightarrow R=\frac{\rho l}{\pi \left(r_2^2-r_1^2\right)}$

Q-3. Three copper wires have their length in the proportion 5:3:1 and their masses are in the proportion  1:3:5. Find the ratio of  electric resistance.

Solution:

$l_1:l_2:l_3:5:3:1\text{ And }{m}_1:m_2:m_3::1:3:5$

$R=\frac{\rho l}{A}=\frac{\rho l}{V}l=\frac{\rho l^2}{m}$

$R\propto \frac{l^2}{m}[\text{ Density }=\text{ Constant }]$

$R_1:R_2:R_3=\frac{25}{1}:\frac{9}{3}:\frac{1}{5}=125:15:1$

Q-4. The resistivity of a copper wire is $1\times 10^{-8}\Omega m$ . If the length of the copper wire is doubled and its area of cross-section is made half, what will be the new value of the resistivity of the wire?

Solution: Since resistivity does not depend upon the physical dimensions i.e, length and area of cross-section of the material, so new resistivity will be unchanged so it remains $1\times 10^{-8}\Omega m.$