Electrical Resistance - JEE Important Topic

Resistance in general language refers to the amount of obstruction or hindrance that blocks someone’s way. In Physics, resistance can come in various forms, such as friction, viscosity, air drag, electrical resistance, thermal resistance and many more. In this section, we will deal with the electrical resistance.

Electrical resistance refers to the resistance that comes into the path of electrons flowing in a current-carrying conductor. All the electrical conductors have some resistance in them. In a current-carrying conductor, when the electrons flow, there are a lot of collisions between the electrons. Such collisions hinder the smooth flow of other electrons, thus, producing resistance.

The resistance can be sometimes beneficial and sometimes detrimental. Resistance in a conductor promotes heat and light. So, a highly resistive electrical component is used to generate heat and light. Since in street light wires or in other wires, a smooth flow of current is required, so, the material with less electrical resistance is taken as a wire material.

Resistance Definition and Origin

Resistance is a measure of the opposition or hindrance offered to the way of flowing electrons. When the electrons flow in a current-carrying conductor, these electrons collide with each other which becomes a hindrance for the other electrons. At the time of collisions, a temporary stoppage comes for some electrons. Then they again start flowing and this process repeats again and again in the conductor.

When the electrons collide, their kinetic energy transforms into heat and light energy. Thus, electrical resistance offers a heating effect. Also, resistance is not only present in current-carrying conductors; the wind also offers resistance called drag, fluids also offer resistance called viscosity and solids offer resistance in the form of friction. Solids also cause thermal resistance.

Resistance Formula

Suppose that a current-carrying wire is connected at the two terminals of the battery providing voltage V. The current of magnitude I flows through the wire. According to Ohm’s law, the voltage is directly proportional to the current, $V\propto I$.

When we remove the proportionality, then a constant of proportionality comes which is called resistance, $V=IR$. This is the formula of resistance. But resistance can also be found by another method.

Suppose that we have a current-carrying conductor of length L and area of cross-section A. The resistance is directly proportional to the length of the conductor and inversely proportional to the area of the cross-section.

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

Removing the proportionality, we will get a constant called resistivity ($\rho $).

Resistivity is the resistance of a current-carrying conductor of unit length and unit area of cross-section. It depends upon the nature of the conductor. So, the resistance formula is:

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

Units of Resistance

• The S.I. unit of resistance is ohm ($\Omega $). But it also has basic units. 

Resistance can be written as $R=\frac{V}{I}$. 

Thus, the units of voltage are volt (V) and the units of current are ampere (A). So, the units of resistance will be V/A.

• As far as the units of resistivity are concerned, the units of length are meter (m), units of area are meter square (m²) and that of resistance is ohm ($\Omega $). So, the units of resistivity will be $\frac{\Omega {{m}^{2}}}{m}=\Omega m$. At the most basic level, the units of resistance are m²kgs^-3A^-2.

Dimensions of Resistance

Now, let us find the dimensional formula of resistance. We know that from Ohm’s Law, resistance R is given by $\dfrac {V}{I}$, and by analysing the formula we can determine the dimensions of the resistance. 

Thus, the dimensions of voltage are $\left [M~{L^2}~T\right]$ and that of current are $\left [{M^0}~{L^0}~{T^0}~I\right]$.

Therefore, the dimensions of the resistance will be:

$\dfrac{\left [M~{L^2}~T\right]}{\left [{M^0}~{L^0}~{T^0}~I\right]}=\left [{M^0}~{L^0}~{T^0}~{I^{-1}}\right]$ .

Hence, the dimensions of resistivity are $\left [{M^0}~{L^0}~{T^0}~{I^{-1}}\right]$.

Electrical Resistance and Conductance

Just like materials have resistance, they have conductance also. Conductance is the opposite of resistance. It is the degree to which a material conducts electricity. Mathematically, conductance is the reciprocal of the resistance. Conductance is denoted by G. The units of conductance are mho or Siemens (S). 

Clearly, the formula of conductance will be $G=\frac{I}{R}$  or

$G=\frac{A}{\rho L}$

$G=\frac{\sigma A}{L} $

Here $\sigma $ denotes conductivity. Conductivity is the reciprocal of resistivity. 

Conductivity is defined as the conductance of a current-carrying conductor having a unit length and unit area of cross-section. The units of conductivity are Siemens per meter (S/m).

Numericals on Resistance

1. A cylindrical current-carrying conductor having a length of 50cm and a radius of 7cm is connected across a battery of 10V. The resistivity of the conductor is $1.72\times {{10}^{-8}}\Omega m$. Calculate the current in the conductor, resistance offered by the conductor, conductance of the conductor and the conductivity of the conductor.

Ans: Given that the length of the conductor is 50cm, the radius of the conductor is 7cm and the resistivity of the conductor is $1.72\times {{10}^{-8}}\Omega m$.

The area of cross-section will be $A=\pi {{r}^{2}}=\frac{22}{7}\times {{\left( 0.07m \right)}^{2}}=0.0154{{m}^{2}}$

The resistance of the conductor is given by

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

$R=\left( 1.72\times {{10}^{-8}}\Omega m \right)\frac{0.5m}{0.0154{{m}^{2}}}$

$R=5.6\times {{10}^{-7}}\Omega$

The current is given by I = V/R

$I=\frac{10V}{5.6\times {{10}^{-7}}\Omega }$ 

$I=1.78\times {{10}^{7}}A$

The conductance is the reciprocal of the resistance.

$G=\frac{1}{R}$

$G=\frac{1}{5.6\times {{10}^{-7}}\Omega }$

$G=1.78\times {{10}^{6}} S $

The conductivity is the reciprocal of the resistivity.

$ \sigma =\frac{1}{\rho }$

$\sigma =\frac{1}{1.72\times {{10}^{-8}}\Omega m}$

$\sigma =5.8\times {{10}^{7}}S/m$

Conclusion

Resistance of a current-carrying conductor is the amount of hindrance offered by the conductor to the flowing current. It is directly proportional to the length of the conductor and inversely proportional to the area of the cross-section of the conductor. The reciprocal of resistance is conductance which is the degree at which the conductor conducts the flow of electrons or current.