Dimesional 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 

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. 

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. 

1. High current densities ​​

These are more likely to favour shorter wavelengths and are more prone to producing continuum emissions.

2. 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].