In the previous chapter we have defined concept of magnetic field represented by vector B
We defined magnetic field B in terms of force it exerts on moving charges and on current carrying conductors
We also know that magnetic field is produced by the motion of the electric charges or electric current
In this chapter we would study the magnetic field produced by the steady current
we would study about how various factors affect the magnitude and direction of the magnetic field
We will also learn to calculate the equation for magnetic field B if the current configuration is known using Biot-savart's law and ampere circuital law
Biot Savart Law
We know that electric current or moving charges are source of magnetic field
A Small current carrying conductor of length dl (length element ) carrying current I is a elementary source of magnetic field .The force on another similar conductor can be expressed conveniently in terms of magnetic field dB due to the first
The dependence of magnetic field dB on current I ,on size and orientation of the length element dl and on distance r was first guessed by Biot and savart
The magnitude of the magnetic field dB at a distance r from a current element dl carrying current I is found to be proportional to I ,to the length dl and inversely proportional to the square of the distance |r|
The direction of the magnetic Field is perpendicular to the line element dl as well as radius r
Mathematically, Field dB is written as
Here (Î¼_{0}/4Ï€) is the proportionality constant such that
Î¼_{0}/4Ï€=10^{-7} Tesla Meter/Ampere(Tm/A)
Figure below illustrates the relation between magnetic field and current element
if in figure, Consider that line element dl and radius vector r connecting line element mid point to the field point P at which field is to be found are in the plane of the paper
From equation (1) ,we expect magnetic field to be perpendicular to both dl and r.Thus direction of dB is the direction of advance of right hand screw whose axis is perpendicular to the plane formed by dl and r and which is rotated from dl to r ( right hand screw rule of vector product)
Thus in figure ,dB at point P is perpendicular directed downwards represented by the symbol (x) and point Q field is directed in upward direction represented by the symbol (â€¢)
The magnitude of magnetic field is
where Î¸ is the angle between the line element dl and radius vector r
The resultant field at point P due to whole conductor can be found by integrating equation (1) over the length of the conductor i.e. B=âˆ«dB
Relation between permeability (Î¼_{0} and permittivity (Îµ_{0}) of the free space
We know that
Î¼_{0}/4Ï€=10^{-7} N/A^{2} ----(a)
and
1/4Ï€Îµ_{0}=9*10^{9} N-m^{2}/C^{2} ----(b)
Dividing equation a by b we get
Î¼_{0}Îµ_{0} =1/(9*10^{16}) (C/Am)^{2}
we know that
1C=1A-s
So Î¼_{0}Îµ_{0} =1/(3*10^{8} m/s)^{2}
And 3*10^{8} m/s is the speed of the light in free space
So Î¼_{0}Îµ_{0} =1/c^{2}
or c=1/âˆš(Î¼_{0}Îµ_{0})
Comparison between Coulomb’s laws and Biot Savart laws
Both the electric and magnetic field depends inversely on square of distance between the source and field point .Both of them are long range forces
Charge element dq producing electric field is a scalar whereas the current element Idl is a vector quantity having direction same as that of flow of current
According to coulomb’s law ,the magnitude of electric field at any point P depends only on the distance of the charge element from any point P .According to Biot savart law ,the direction of magnetic field is perpendicular to the current element as well as to the line joining the current element to the point P
Both electric field and magnetic field are proportional to the source strength namely charge and current element respectively. This linearity makes it simple to find the field due to more complicated distribution of charge and current by superposing those due to elementary changes and current elements