Dimension of Resistance

Dimensions

Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. 


Dimensional Formula

The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e ( ).

E.g: Dimension formula of mass is: (M)


Dimensional Equation 

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. 


Application of Dimensional Analysis

1. To convert a physical quantity from one system of the unit to the other:

It is based on a fact that magnitude of a physical quantity remain same whatever system is used for measurement i.e magnitude = numeric value(n) multiplied by unit (u) = constant

n1u1= n1u2

2. To check dimensional correctness of a given physical relation: 

If in a given relation, the terms of both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions. 

3. To derive a relationship between different physical quantities: 

Using the principle of homogeneity of dimension, the new relation among physical quantities can be derived if the dependent quantities are known. 


Limitation of this Method

1. This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric and logarithmic functions can not be derived using this method. The formula containing more than one term which is added or subtracted likes s = ut+ ½ at2 also cannot be derived.

2. The relation derived from this method gives no information about the dimensionless constants. 


Dimension of Resistance

Dimension formula of resistance is: length2 × mass × time-3 × electric-current-2 (M1 L2 T-3 I-2)

Where,

  • M = Mass

  • I = Current

  • L = Length

  • T = Time


Derivation of Resistance Dimension

Resistance (R) = Voltage × Current-1 . . . . (1)

As we all know the formula of voltage (V) = Electric Field × Distance = [Force × Charge-1] × Distance

The dimension formula of force can be written as M1 L1 T-2

The dimensional formula of charge = current × time = I1 T1

∴ The dimensional formula of voltage is written as [Force × Charge-1] × Distance

= [M1 L1 T-2] × [I1 T1]-1 × [L1] = [M1 L2 T-3 I-1] . . . . (2)

On placing equation (2) in equation (1) we get,

Resistance (R) = Voltage × Current-1

Or, R = [M1 L2 T-3 I-1] × [I]-1 = [M1 L2 T-3 I-2]

Therefore, the resistance is dimensionally written as M L2 T-3 I-2.


Define Resistance

The resistance of the conductor is the opposition offered by a conductor during the flow of change. When the potential difference is applied across the conductor, free electrons get accelerated and collide with positive ions and their motion is thus opposed. This opposition offered by ions is called resistance of the conductor. Hence the resistance is the property of the conductor by which it opposes the flow of current in it. 

Unit of resistance is the ohm.

Factors on which Resistance is Dependent

1. Length of the conductor.

2. Area of a cross-section of the conductor.

3. It is dependent on the material of the conductor but does not depend on the geometry of the conductor.

4. The resistance of the conductor depends on the temperature of the conductor.


Resistivity

The specific resistance of the material is equal to the resistance of the wire of that material unit cross-section area and unit digit. 

Factors on which Resistivity Depends

1.  Nature of material.

2. Temperature of material.

FAQ (Frequently Asked Questions)

1. Define Dimension Formula.

Ans. Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e ( ).

2. Explain a Few Limitations of Dimension Formulas.

Ans. Some of the limitations of dimension formula are given below:

  • This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric and logarithmic functions can not be derived using this method. A formula containing more than one term which is added or subtracted likes s = ut+ ½ at2 also cannot be derived.

  • The relation derived from this method gives no information about the dimensionless constants.

3. Write a few sets that have the same Dimension Formula.

Ans. Some of the sets having the same dimension formula that are discussed below:

  • Strain, refractive index, relative density, angle, solid angle, phase, distance gradient, relative permeability, angle of content.

  • Mass and inertia.

  • Momentum and impulse.

  • Thrust, force, weight, tension, energy gradient.

  • Angular momentum and plaques contant.

  • Surface tension, surface area, force gradient, spring constant.

  • Latent heat and gravitational potential.

  • Thermal capacity, Boltzman constant, entropy.