Find magnetic field at larger arc

Question

battery is connected between point X and y on the circumference of uniform conducting ring of radius a and resistance R .The smaller arc betweenX and y is substance and angle 60 degree at the centre and magnetic induction at due to current in the small arc is 0.1 Tesla in the outward  direction magnetic induction at due to current in the larger arc is between x and y is

1)0.6T  in the inward direction.

2)0.1 T in the inward direction

3)0.6 T in the outward direction.

4) 0.1 T in the outward direction.

Answer 1

Correct option(2)

Explanation:

Let us consider the magnetic field due to current flowing in anticlockwise direction in a circular arc of radius R and  length L as shown in figure 1.   

Magnetic field B at centre O is given by Biot-savarts  law ,  B =

Tesla  

Hence magnetic field is proportional to arc Length and Current. 

If current flows in anti-clockwise direction magnetic field is outward. 

If current flows in clockwise direction magnetic field is inward.   

In Figure.2, if  anti-clockwise direction current flowing in minor arc produces 0.1T magnetic field at O in outward direction, 

Magnitude of Magnetic field at O due to clockwise direction current flowing in major arc depends on the product of current and arc length.   

Since arc length of major arc is 5 times of arc length of minor arc, its resistance increases by 5 times ,   hence current in major arc becomes (1/5) of current flowing in minor arc.  

Hence the product of ( current × arc Length ) remains same, because current becomes 1/5 and length becomes 5 times. 

Hence magnitude of magnetic field will be same. 

But due to change in current direction, direction of magnetic field due to major arc is inward.