Magnetic Field Due To Circular Current Loop - Practice Questions & MCQ

Magnetic Field due to circular current loop at its centre: 

Magnetic Field due to circular coil at Centre-

Consider a circular coil of radius a and carrying current  I  in the direction shown in Figure. Suppose the loop lies in the plane of the paper. It is desired to find the magnetic field at the centre  O of the coil. Suppose the entire circular coil is divided into a large number of current elements, each of length dl.

According to Biot-Savart law, the magnetic field  at the centre  O of the coil due to the current element    is given by,

where is the position vector of point O from the current element.The magnitude of    at the centre  O  is

 

 

The direction of    is perpendicular to the plane of the coil and is directed inwards. Since each current element contributes to the magnetic field in the same direction, the total magnetic field  B  at the centre O  can be found by integrating the above equation around the loop i.e.

For each current element, angle between $\overrightarrow{\mathrm{dI}}$ and $\vec{r}$ is $90^{\circ}$. Also distance of each current element from the center O  is a.

  

But 

 

For N turns,

 

where N=number of turns, i= current and r=radius of a circular coil.

 

Magnetic field due to a current-carrying circular arc

Case 1:  Arc subtends angle theta at the centre as shown below then  

Proof:

                           

Consider length element dl lying always perpendicular to .

Using the Biot-Savart law, the magnetic field produced at O is:

                                                                                  

 

Equation (1) gives the magnitude of the field. The direction of the field is given by the right-hand rule. Thus, the direction of each of the dB is into the plane of the paper. The total field at O is

The angle subtended by element is at pt. O, therefore
                                                                       

where the angle  is in radians.

Case 2:  Arc subtends angle () at the centre then 

Case 3:The magnetic field of the Semicircular arc at the centre  is 

Case 4: Magnetic field due to three-quarter Semicircular Current-Carrying arc at the centre 

 

Special cases 

1. If the  Distribution of current across the diameter then 

2. If Current between any two points on the circumference then 

 

3. Concentric co-planar circular loops carrying the same current in the Same Direction-

 

  

If the direction of currents are the same in concentric circles but have a different number of turns then 

 

 

4. Concentric co-planar circular loops carrying the same current in the opposite Direction 

 

 

If the number of turns is not the same i.e 

5.  Concentric loops but their planes are perpendicular to each other 

Then 

 

6. Concentric loops but their planes are at an angle ϴ with each other