(a) Using Biot-Savart’s law, derive the expression for the magnetic field in the vector form at a point on the axis of a circular current loop.

(b) What does a toroid consist of? Find out the expression for the magnetic field inside a toroid for N turns of the coil having the average radius r and carrying a current I. Show that the magnetic field in the open space inside and exterior to the toroid is zero.

Biot-Savart law states that the magnetic field strength (dB) produced due to a current element of current I and length dl at a point having position vector  to current element is given by, 
                       

where,  is permeability of free space.

 The magnitude of magnetic field is given by,

  ; is the angle between the current element and position vector. 

Magnetic field at the axis of a circular loop:

Consider a circular loop of radius R carrying current I. Let, P be a point on the axis of the circular loop at a distance x from its centre O. Let,  be a small current element at point A.

Magnitude of magnetic induction dB at point P due to this current element is given by,
                                       ... (1) 
The direction of  is perpendicular to the plane containing .
Angle between  
Therefore, 
                     ... (2) 

The magnetic induction  can also be resolved into two components, PM and PN’ along the axis and perpendicular to the axis respectively. Thus if we consider the magnetic induction produced by the whole of the circular coil, then by symmetry the components of magnetic induction perpendicular to the axis will be cancelled out, while those parallel to the axis will be added up. 

Thus, resultant magnetic induction  at axial point P is given by, 



 

Therefore the magnitude of resultant magnetic induction at axial point P due to the whole circular coil is given by, 
B = 

Therefore, 

b) A long solenoid on bending in the form of closed ring is called a toroidal solenoid.

i) For points inside the core of a toroid,

 As per Ampere’s circuital law, 
 
where, I is the current in the solenoid.

 So, resultant net current = NI
 

Since no current is flowing through the points in the open space inside the toroid.

Therefore, I = 0.
So,