Statement
The magnetic field at a point due to a current element of a conductor is (i) directly proportional to the current, (ii) directly proportional to the length of the current element, (iii) directly proportional to the sine of the angle between the element and the line Joining the point to the mid point of the element and (iv) inversely proportional to the square of the distance of the point from the element.
Consider a small element AB of length dl of a conductor XY carrying a current i [Figure given below]. Let \vec r be the position vector of the point P from the current element i\vec{dl} and \theta be the angle between \vec {dl} and \vec r. Current element i\vec{dl} is a vector, which is the product of the current i and length dl of very small segment of the current carrying conductor. Its length is dl and direction is the tangent to the element and is acting in the direction of current flowing through the conductor. According to Biot Savart law, the magnetic field dB at the point P due to the current element depends as
(i)dB\propto i ; dB\propto dl ; dB\propto \sin\theta ; dB\propto \frac1{r^2}
Combining,
dB\propto \frac{idl\sin\theta}{r^2}
or, dB=\kappa\frac{idl\sin\theta}{r^2}
Where – is a constant of proportionality, whose value depends on the system of unit chosen and medium between the point P and the current element.
Where \kappa is a constant of proportionality, whose value depends on the system of unit chosen and medium between the point P and the current element.
In SI,
\kappa = \frac{\mu_0}{4\pi} in vacuum
And in CGS
\kappa = 1 in vacuum
Where \mu_0 is the absolute permeability in vacuum.
\mu_0= 4\pi\times {10}^{-7} Wb A^{-1}m = 4\pi \times {10}^{-7}TA^{-1}m
so, dB = \frac{\mu_0}{4\pi}\frac{idl\sin\theta}{r^2} in SI
and dB = \frac{idl\sin\theta}{r^2} in CGS
Biot Savart Law in Vector form
In vector form,
\vec {dB} = \frac{\mu_0}{4\pi}\frac{i(\vec{dl}\times \vec{r})}{r^3} in SI
and \vec{dB}=\frac{i(\vec{dl}\times \vec r)}{r^3} in CGS
The direction of \vec{dB} is the direction of the cross product (\vec{dl}\times \vec r). It is obtained by the right hand
screw law of vector product. \vec {dB} is perpendicular to the plane containing \vec{dl} and \vec r and is directed inwards. If the point P be to the left of the current element, \vec {dB} will be directed outwards.
Magnetic field induction at a point P due to current through entire wire is
\vec B= \frac{\mu_0}{4\pi}\int \frac{I \vec{dl} \times\vec r}{r^3}
B = \frac{\mu_0}{4\pi}\int\frac{I dl \sin\theta}{r^2}
Biot Savart law in terms of current density
\vec{dB} = \frac{\mu_0}{4\pi}\frac{\vec J\times \vec r}{r^3}dV
[ \therefore J= \frac{I}{A}= \frac{I dl}{A dl}=\frac{I dl}{dV} ]
where J = current dnsity at any point on thecurrent element, dV = volume of the element.
Biot Savart law in terms of charge (q) and velocity (v)
\vec{dB}= \frac{\mu_0}{4\pi}\frac{q(\vec{dl} \times \vec{r})}{r^3}
\because I\vec{dl} = \frac{q}{dt}.dl = q \frac{\vec {dl} }{dt} = q\vec v
Biot Savart Law in terms of magnetisingforce or magnetising intensity (H) of the magnetic field
\vec{dH}= \frac{\vec{dB} }{\mu_0} = frac1{4\pi} \frac{I\vec{dl} \times \vec r}{r^3}
and dH = \frac1{4\pi} \frac{I dl\sin\theta}{r^2} in SI system.
\vec{dH}= \frac{I \vec{dl} \times \vec r}{r^3}
and dH = \frac{I dl \sin\theta}{r^2} in CGS system.
