Dimension of Resistance: Dimensional Formula and Equation, Derivation, Applications, and Limitations

Dimension of Resistance Overview

Resistance is a measure of the resistance to electron flow and is computed by dividing voltage by the current in the wire. The symbol for the unit of resistance is Ohm (Ω).  As an electric current travels through a bulb, the bulb creates certain impediments to the current flow, which is known as electrical resistance. The opposition formed by a material against the passage of electricity is known as resistance. 

What is Resistance?

The conductor's resistance is the opposition provided by a conductor during the passage of change. When a potential difference is placed across a conductor, free electrons are accelerated and collide with positive ions, causing their motions to be opposed. This opposition provided by ions is referred to as conductor resistance. So, resistance is the characteristic of a conductor that opposes the flow of current through it.

Dimensional Formula of Resistance

Any physical quantity's dimensional formula is the statement that represents how and which of the basic quantities are included in that quantity. It is expressed by encircling the symbols for base amounts with the appropriate power in square brackets, for example ( ).

For example, the mass dimension formula is (M).

Importance of the Dimension of Resistance

One of the most essential subjects included in the IIT JEE syllabus is "Dimension of Resistance." Knowing more about resistance and the dimensional formula of resistance can improve their understanding of the physics topic, which accounts for a significant chunk of the IIT JEE test question paper. One will be able to answer the practical problems in your IIT JEE test with ease if one grasps the dimensional formula of resistance and its applications.

What is a Dimensional Equation?

A "dimensional equation" is an equation derived by equating a physical quantity with its dimensional formula.

Dimensional Formula of Resistance

The resistance dimensional formula is provided by-

M¹L²T⁻³I⁻²

Where,

M = Mass 

L = Length

T = Time

I = Current

How to Calculate the Dimensional Formula of Resistance?

The dimensional formula of resistance may be calculated using Ohm's law. Ohm's law's mathematical formula is-

Voltage (V) = Current (I) * Resistance (R)

Resistance (R) = Voltage (V) / Current (I)

Let us now deconstruct this formula in terms of fundamental quantities-

Voltage = Work / Charge

By substituting Force * displacement for the amount of Work,

Voltage = (Force * Displacement) / Charge

By substituting Mass * Acceleration for the value of Force,

Voltage = (Mass * Acceleration  * Displacement) / Charge

Also,

Voltage = (mass * distance) / time² * displacement) / charge

               = (mass * distance * displacement) / (charge * time²)

Now the formula of Resistance becomes, 

Resistance = (mass * distance * displacement) / (Charge * Current * time²)

Also, the formula for charge is "current * time."

Hence, the formula becomes,

Resistance = (mass * distance * displacement) / (Current * time * Current * time²)

= (mass * distance * displacement) / (Current² * time³)

= (mass * distance * displacement * time⁻³ * Current⁻²)

Resistance = [M¹] * [L²] * [T⁻³] * [A⁻²]

Hence, the dimensional equation of resistance is R = [M¹ L² T⁻³ A⁻²].

Limitations of Dimension of Resistance

This approach can only be used if the dependence is of the multiplication type. This approach cannot be used to create formulas involving exponential, trigonometric, or logarithmic functions. A formula having more than one component that is added or removed, such as s = ut + 12 at2, cannot also be derived. The dimensionless constants are not mentioned in the relation obtained using this technique.

Popular Resistance Units 

The following table contains some most popular resistance units-

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Applications of Dimension of Resistance 

The following are some of the applications of dimensions of resistance-

To convert a physical quantity from one unit system to another- 

It is based on the fact that the magnitude of a physical amount remains constant regardless of the measuring method employed, i.e.,

magnitude = numeric value(n) multiplied by unit (u) = constant

To determine the dimensional accuracy of a particular physical relationship-

If the terms on both sides of a given relation have the same dimensions, then the equation is dimensionally accurate. This notion is best known as the principle of dimension homogeneity.

To establish a link between several physical quantities-

If the dependent values are known, the new connection among physical quantities may be determined using the concept of dimension homogeneity.

Factors on which Dimension of Resistance Depends

The following are some of the factors on which resistance depends-

1. It is affected by the conductor's substance but is unaffected by the conductor's geometry.
2. The conductor's cross-sectional area.
3. The conductor's length.
4. The resistance of the conductor is affected by its temperature.

Conclusion

Resistors are the essential components of every circuit commonly utilized to limit the quantity of current flow. Before employing resistors, it is important to understand their features so that they may be utilized particularly for a given demand and kind of circuit. The current flow is the result of the resistance and potential difference operating across the two ends of the circuit. The dimensional analysis of resistance allows for the discovery of the fundamental units of the quantity, paving the way for more accurate and smooth computations and comparisons with other physical quantities.