About Dimensional Formula of Current Density
Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. Dimensions of any given quantity tell us about how and which way different physical quantities are related. Finding dimensions of different physical quantities has many real-life applications and is helpful in finding units and measurements. Imagine a physical quantity X which depends mainly on base mass(m), length(L), and time(T) with their respective powers, then we can represent the dimensional formula as [MaLbTc].
Dimensional Formula
The dimensional formula of any physical quantity is that expression that 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
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
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.
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
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.
The relation derived from this method gives no information about the dimensionless constants.
Electric Current
When charge flows in a conductor from one place to another, then the rate of flow of charge is called electric current. When there is a transfer of charge from one point to another point in a conductor, we say that there is electric current through the area. If the moving charge is positive, the current is in the direction of charge. If moving charges are negative the current is opposite to the direction of motion of charge.
If a charge ∂Q crosses an area in time ∂t, we define average electric current through the area, during this time as
Average current I(average) = \[\frac{\partial Q}{\partial t}\]
Instantaneous current = \[\frac{dQ}{dt}\]
Unit of current is ampere (A) or coulomb/sec
i.e Q= ne therefore I=ne/t
Here n= number of free electrons
t= time in which n electron passes through it
e= charge on electron
Properties of Electric Current
The flow of electrons through a conductor is called electricity and an electric current tends to generate heat energy.
An electric current is referred to as the flow of charge per unit time.
There is a relation between the amount of current and the voltage that pushes the electrons, along with the degree of resistance to flow.
The SI unit of electric current is ampere. It is denoted by ‘A’.
The longer the current flows through a conductor, the more the heat is generated.
Current Density
The current density at a point in a conductor is the ratio of the current at that point in the conductor to the area of cross-section of the conductor of that point.
\[\text{Unit}\] = \[\frac{ \text {ampere}} { \text {metre}}\].
\[j=\frac{dI}{dA}\], here dA is the cross-section area
Application of Current Density
Current density plays a vital role in the outward spectrum that is produced in gas discharge lamps, like flash lamps.
High current densities
These are more likely to favour shorter wavelengths and are more prone to producing continuum emissions.
Low current densities
These, on the other hand, are more likely to favour longer wavelengths and are more prone to producing spectral line emissions.
The Dimension of Current Density
The dimensional formula of current density is given by, M0 L-2 T0 I1
Where, standard unit mass is represented as M, current by I, length by L, and time by T.
Increase and Decrease of Current Density
The current density will be higher in a conductor if the current in it is high. But the current density tends to alter in different parts of an electric conductor. And the effect tends to take place with alternating currents at higher frequencies.
When the charging current increases, the time that is required to build up the same charge/voltage tends to decrease proportionally. Therefore, the charging-discharging time cycle tends to decrease proportional to the current, when the current increases.
Derivation of the Dimensional Formula of Current Density
Formula of Current Density = Charge ×(Area × Time1) . . . (1)
As charge = current × time
Therefore, the dimensions of electric charge can be written as M0L0I1T1 . . . . (2)
And, the dimensional formula of some required units are given below:
Area = [M0 L2 T0] . . . (3)
Time = [M0 L0 T1] . . . . (4)
On putting equation (2) and (3) in equation (1) we get,
Current Density = Charge × Area×Time-1
= [M0 L0 I1 T1] × [M0 L2 T0]-1 × [M0 L0 T1]-1 = [M0 L-2 T0 I1].
Therefore, the dimensional formula of current density is represented as [M0 L-2 T0 I1].
FAQs on Dimesional Formula of Current Density
1. Define Dimension Formula.
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.
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 dimensional formula.
Some of the sets having the same dimension formula that are discussed below:
Strain, refractive index, relative density, distance gradient, relative permeability, angle of content.
Mass and inertia.
Momentum and impulse.
Thrust, force, weight, tension, energy gradient.
Angular momentum and Planck’s constant.
Surface tension, surface area, force gradient, spring constant.
Latent heat and gravitational potential.
Thermal capacity, Boltzman constant, entropy.
4. What is the dimensional formula? List its uses along with its limitations.
The expression that indicates how and which of the base quantities are included in a given physical quantity is known as its dimensional formula. And the equation that you get when you equate the physical quantity along with its dimensional formula, is known as a dimensional equation.
The following are the advantages/uses of the dimensional formula:
It can be used to convert the physical quantity from one system of the unit to another.
It can be used to check the dimensional correctness of physical relations.
It can be used to derive a relationship between various physical quantities.
The formula has a few limitations:
It can only be utilised if the dependency is of multiplication type.
The relation that we get through this method gives us no sort of information about the constants that are dimensionless.
5. Discuss the application of dimensional analysis.
The application of dimensional analysis is as follows:
Converting a physical quantity from one unit to another: This is dependent on the fact that the physical quantity’s magnitude remains constant regardless of which system is used for the measurement.
Checking the dimensional correctness of a physical relation: This concept is also popularly known as the principle of homogeneity of dimensions. It claims that an equation is dimensionally correct if, in a particular equation, the terms present on both sides have the same dimensions.
Deriving a relation between different physical quantities: The new relation amongst the physical quantities can be derived using the same concept as mentioned above, given the dependent quantities are known.
6. Explain the process of electric current.
When charge flows through a conductor from one point to another, the rate of its flow is known as electric current. If that moving charge is positive, then the current flows in the direction of the charge. And if the moving charge is negative, then the current tends to flow in the direction opposite from the direction of motion of the charge. Ampere or amp is considered to be the SI unit of electric current, i.e., the flow of electric charge across the given surface at the rate of one coulomb/second.
7. Explain current density and the dimension of current density.
In a conductor, the current density at a given point is referred to as the ratio of the current at that particular point, to the area of cross-section of that point. Its unit is ampere/metre. Current density is fundamental to the design of electronic systems.
The dimensional formula of current density is as follows:
[M0 L-2 T0 I1], where M stands for the standard unit mass, I stands for current, L indicates the length, and T represents the time.
