Resistance And Resistivity - Practice Questions & MCQ

Resistance

• The resistance is known as the property of substance by virtue of which it opposes the flow of current through it.

• Formula-

For a conductor of resistivity $\rho$ having a length of a conductor= l

and  Area of a crosssection of conductor= A

Then the resistance of a conductor is given as 

$
R=\rho \frac{l}{A}
$

Where $\rho \rightarrow$ Resistivity

•  Its S.I unit is Volt/Amp or ohm $(\Omega)$
•  Its Dimensions is $M L^2 T^{-3} A^{-2}$

• Reciprocal of resistance is known as conductance.

• Resistance of a conductor depends on the following factors

1.  Length -

As $R=\rho \frac{l}{A}$
So Resistance of a conductor is directly proportional to its length
i.e. $R \alpha l$
2. Area of cross-section-

As $R=\rho \frac{l}{A}$
Resistance of a conductor is inversely proportional to its area of cross-section
i.e. $R \alpha \frac{1}{A}$
3. Material of the conductor-

As $R=\rho \frac{l}{A}$

And For a conductor, if $n=$ No. of free electrons per unit volume in the conductor, $\tau=$ relaxation time then the resistance of conductor
${ }_{\text {Then }} \rho=\frac{m}{n e^2 \tau}$
for different conductors n is different
And $\rho$ depends on n
So R is also different.
4. Temperature-
${ }_{\text {As }} R=\rho \frac{l}{A}$
And $\rho=\frac{m}{n e^2 \tau}$
So $R \propto \frac{1}{\tau}$

And as temperature increases decrease

So as the temperature increases resistance increases

 Temperature-dependent  resistance is given by

$
R_T=R_{T_0}\left[1+\alpha\left[T-T_0\right]\right]
$

$R_T$ - Resistance at temperature $T$
$R_0$ - Resistance at temperature $T_o$
$\alpha$ - temperature coefficient of resistance

$
\alpha=\frac{R_T-R_o}{R_o\left(T-T_o\right)}
$

Where the value of $\alpha$ is different at different temperatures
- From Ohm's law

$
V=I R
$

                   Where R- Electric Resistance

1. Ohmic Substance: The substance which obeys Ohm's law are known as Ohmic substance. I-V graph is linear and the slope gives conductance which is reciprocal of resistance

2.   Non-ohmic substances

Those substances which don't obey Ohm's law are known as Non-ohmic or non-linear conductors.

For example gases, crystal rectifiers, etc. 

3.    Superconductor: For certain materials resistivity suddenly becomes zero below a certain temperature (critical temperature). The material in this state is called a superconductor.

In Superconductor, resistivity is zero

• Resistivity or Specific Resistance $(\rho)$

- As

$
R=\rho \frac{l}{A}
$

If $\mathrm{l}=1 \mathrm{~m}$ and $\mathrm{A}=1 \mathrm{~m}^{\mathrm{A}} 2$
Then $\mathbf{R}=\rho$

Resistivity is numerically equal to the resistance of a substance having a unit area of cross-section and unit length.
- Where m is the mass, n is the number of electrons per unit volume, e is the charge of electron and $\tau$ is the relaxation time

Then $\rho=\frac{m}{n e^2 \tau}$
- S.IUnit - Ohm.m
- Dimensions- $M L^3 T^{-3} A^{-2}$

And as reciprocal of Resistivity is known as conductivity.
So the dimension of conductivity is $M^{-1} L^{-3} T^3 A^2$

•  Resistivity is independent of the shape and size of the body as it is an intrinsic property of the substance.

The resistivity of a conductor depends on the following factors

1. Nature of the body-

$
{ }_{\mathrm{As}} \rho=\frac{m}{n e^2 \tau}
$

for different conductors n is different
And $\rho$ depends on $n$
So $\rho$ is also different.

Temperature-dependent Resistivity :

$
\rho=\rho_o\left(1+\alpha\left(T-T_o\right)\right)
$

$\rho$ : Resistivity at temperature T
$\rho_0$ : Resistivity at the temperature $T_0$

• Resistivity increases with impurity and mechanical stress.