Biot-Savart’s law: It states that the magnetic field induction dB at point P due to a current element \(I \vec{dl}\) is given as
(i) dB ∝ I
(ii) dB ∝ dl
(iii) dB ∝ sin θ
(iv) dB ∝ \(\frac{1}{r^2}\)
Combining these factors, we get
dB ∝ \(\)= \(\frac{Idl\ sin\ \theta}{r^2} = \frac{\mu_0}{4\pi} \frac{Idle\ sin\ \theta}{r^2}\)
In vector form
\(d \vec B = \) \(\frac{\mu_0}{4\pi}I \frac{\vec {dl} \times \vec r}{r^3}\)
The direction of \(d \vec B\) is perpendicular to the plane of paper and directed inwards.
Intensity of magnetic field at the centre of a circular coil carrying current. Consider a conductor bent into the form of the loop of a circle and carrying current I as shown in Fig. Let us determine the intensity of the magnetic field at the centre of the circle O.
Consider an element AB = \(d\ \vec l\) of the circular loop. The magnetic field at the centre O due to this current element is given by
\(\vec {dB} = \) \(\frac{\mu_0}{4\pi} .\frac{I\vec {dl} \times \vec r}{r^3}\)
where \(\vec r\) is the vector from current element of length \(\vec {dl}\) to the point O.
Since the angle θ between dl and r is 90°, we have
dB = \(\frac{\mu_0}{4\pi}.\frac{Idl}{r^2}\)............(1)
The direction of this field is perpendicular to the plane of the loop and directed away from the reader and is shown by the sign of a cross in a circle (resembling the tail of an arrow).
The loop can be divided into a large number of such current elements. The magnitude of the field due to each current element is the same as that given by Eqn. (1) and is perpendicular to the plane of the loop directed away from the reader.
The total magnetic field at O is given by
If the coil consists of n turns wound one over the other, we have
B = \(\frac{\mu_0nI}{2r}\) tesla ...........(3)
Magnetic field lines of circular current loop
The magnetic field lines due to circular wire form a closed loop are shown in Fig. and the direction of magnetic field is given by right hand thumb rule.
A current carrying circular loop has magnetic field around it and its magnetic moment is IA, hence it behaves like a magnet.
