Dimensional Formula of Current Density

In electromagnetism, the current density is defined as the electric current passing through a substance per unit time in an area of cross-section. It is a vector quantity, meaning it has both magnitude and direction. It affects the power generated by electronic devices. The unit we can measure current density is ampere per metre square.

Current Density and Drift Velocity

The current density is the quantity of charge per unit time flowing in a unit cross-section of a conductive material. The relation between current density and drift velocity is formalised as:

J = I/A.

Here, ‘J’ is the current density. 

We know that drift velocity is calculated with the formula:

I = nAvdQ 

where,

‘I’ is the charge passing through a conductive material,

‘n’ is the total number of charged particles or electrons,

‘A’ is the cross-sectional area of the material,

‘vd’ is the drift velocity of the charged particles in the material, and

‘Q’ is the electron charge in Coulombs.

Thus, substituting the values for ‘I’ in current density, we get the following result:

J = nvdQ.

In other words, the drift velocity is directly proportional to the current density of the charged particles. Furthermore, the velocity grows on increasing the force of the electric field, resulting in a significant increase in the amount of current passing through the conductive material.

Dimensional Formula of Current density

The formula of current density is given by, j = charge * (area * time-1) 

We know that, charge = current * time

So, charge = [M0L0I1T1] , Area = [M0L2T0] and time = [M0L0T-1] 

By adding all these dimensional formulas in the current density we get,

j =  [M0L0I1T1] * [M0L2T0] * [M0L2T-1]-1

  = [M0L-2T0I1]

Here, M = mass, L = length, T = time and I = current

Therefore, the dimensional formula of current density is given by  [M0L-2T0I1]

Dimensional formula of Current density importance:

• The dimensional formula of current density helps us understand the physical correctness of any equation involving force.
• It helps us understand the relationship between different physical quantities involving current density.
• It helps us in converting units from one physical quantity to another.
• In any relationship, this analysis can find the constant dimensions.

The formula for Drift Velocity

Net Velocity of Electron: The electrons that remain present in the conductors usually move randomly without any proper direction, resulting in collisions. The current flow can occur when they will not collide and move in a particular direction systematically. However, once they hit, the Net velocity of the electrons will perform the same. 

For instance, consider the first collision of electrons to be T1, and then the second set of Electrons T2 will do the same and continue. Apart from this, if you consider the overall Net velocity of the electrons to be n that includes T1 – Tn, then the Formula for Drift Velocity in respect of Relaxation time is  (T) = (T1+T2+…Tn)/n

Also, we know that V = U + aT

Thus, we get that ‘V’ represents the Net velocity of the electrons,

‘U’ is for the starting Velocity,

‘a’ is for Acceleration, and lastly, 

‘T’ is the Time. 

Similarly, V = aT, and acceleration is a = F/m = -qE/m 

Then the Drift Velocity =  Vd = – qEmT

However, another Formula for drift velocity provides the relation between the current and drift velocity, including I =  nAvq. 

Dimensional Formula

The physical quantity’s dimensions are the powers to which the basic quantities are elevated to represent that amount. The dimensional formula of any physical quantity is an equation that explains how and which of the base quantities are contained in that amount. It is written by enclosing the symbols representing base amounts in square brackets with the corresponding power, i.e. [].

E.g.: the dimension formula of displacement is: [L]

A dimensional equation is obtained by equating a physical quantity with its dimensional formula is used to calculate the dimensional equation of any physical quantity.

Current density importance

Current density plays a significant role in designing different electronic and electrical equipment. A lot of complicated distributions in terms of current, charge can be understood through current density equations, and it also impacts the reactance and resistance values.

Conclusion

Current density is a widely used term in electromagnetism and can be defined as the amount of electric current in unit time in a cross-sectional area of a material. It is used in designing electronic and electrical equipment, affecting the power generated in a device. The unit to measure it is amperes per metre square, and its dimensional formula is given by [M0L-2T0I1].