Dimensional Formula of Resistance

The single directional movement of the electrons in any material is termed current. The materials with free electrons do not flow because of random equal directional movement of the electrons; only a directional drift can cause current to flow in any material. 

Resistance is termed as the hindrance to the flow of electrons or the opposed movement of electrons that occurred due to potential differences across the conductor. The combination of opposing and encouraging forces makes the current flow in a circuit. Its SI unit is ohm and represented by Ω.

What is Electrical Resistance?

The sub-atomic configuration of any material reveals that it has free electrons that are in random movement. The application of potential differences across any material leads to the alignment of these charge carriers. Depending on the majority of the charge carriers, the direction of flow of current is established. However, it is the property of the material to restrict the flow of current that disturbs its original state. Hence, the material opposes the current.

A relation is observed between the potential difference applied across the conductor and the current passing through it. Let V be the potential difference, and I be the current in the following circuit.

Then, resistance R can be expressed by:

V ∝ I 

V = IR

V is expressed in volts

I is expressed in amperes

R is expressed in ohms

In terms of electrical resistance:

R = V/I

The SI unit of electrical resistance is  ohm. Hence, it is the measure of the easiness of the electric current passing through the conductor. The different materials have different resistance depending on multiple factors.

Dimensional Formula:

The representation of a fundamental unit of a physical quantity is expressed in the form of [MaLbTc], where M is the mass, L is the length, and T stands for  time where a,b and c are exponents.

Dimensional Analysis and its application:

Dimensional analysis is the process of finding the relationship between two physical quantities, where the quantities are the group of fundamental units, independent of numerical values and constant. The idea of similar units makes the comparison and calculations smooth.

Application of dimensional analysis:

• Conversion of the fundamental unit of one quantity to another.
• Monitoring over the accuracy of the relationship between two quantities.
• To obtain a relationship between two different physical quantities.

Dimensional Formula of Resistance:

The opposing force to the current flow is stated as the resistance in an electric circuit. The dimensional formula representation of the resistance is expressed as:

[M1L2T-3I-2]

Where 

M: Mass 

L: Length

T: Time 

I: Current

Derivation of dimensional representation of resistance:

The relation between the potential difference and resistance results in the flow of electrons, the resultant flow of current is the drift of electrons from high concentration to low concentration due to existential potential difference and the resistivity.

• According to Ohm’s Law

V= (I).(R) or R= (V) / (I)

Where,

R: Resistance

V: Voltage

I: Current

The dimension of current is [I1]

• Formula of Potential difference (voltage V = (E).(d) or E = (F) / (q))

Therefore, the dimension of force is [M1L1T−2]

The dimension of charge is [I1T1]

Therefore dimension of E = [M1L1T−2] / [I1T1] = [M1L1T−3I−1]

The dimension of V = [M1L1T−3I−1] . [L] = [M1L2T−3I−1]

So, the dimensional formula of resistance  [M1L2T−3I−1] / [I1] = [M1L2T-3I-2]

Dimensional homogeneity principle: 

The notion of dimension homogeneity requires that all words in a physical statement have the same dimensions. In the physical formula v2 = u2 + 2as, for example, the dimensions of v2, u2, and 2as are identical and equal to [L2T-2].

Factor labelling using dimensional analysis:

It is possible to convert units between different measurement systems. Factor label technique, unit factor method, or dimensional analysis are used to describe this approach.

Conversion factors are used to express the connection between different system’s units, making it possible to move an item from one to another. Based on the fact that the ratio of each basic quantity in one unit with their counterpart in another unit is equal to one.

Conclusion:

The resistors are the basic components of any circuit which is widely used to limit the amount of flow of current. Before using the resistors, the characteristics of the resistor should be well acknowledged so that it can be used specifically as per need and a particular type of circuit. The flow of current is the resultant of the combination of the resistance and potential difference acting across the two ends of the circuit. The dimensional analysis of the resistance allows obtaining knowledge about the fundamental units of the quantity, which leads the way for easier and smooth calculations and comparisons with any other physical quantities.