Question

Two concentrated co-planar coils of equal turn have radii $10$ cm and $30$ cm respectively. The same current is passed into in both the coils in the same direction. Now the direction of currents is reversed in one coil then, the ratio of magnetic flux at their common centre is:

• A. 5:4
  
• B. 4:3
  
• C. 1:1
• D. 1:2
  

Answer 1

The correct option is A 5:4

Magnetic field due to one coil is :

$B=\frac{{\mu }_{o}NI}{L}$ [N = No of turns]

When current flows in same direction, they produce magnetic field in same direction, so flux :-

$\Rightarrow {\varphi }_2=\frac{{\mu }_{o}NI}{L}\pi (R_1^2+R_2^2)$

But when current flows in apposite direction, the magnetic field of one coil will be in reverse, so flux,

${\varphi }_2=\frac{{\mu }_{o}NI}{L}\pi (R_1^2-R_2^2)$ [Given $R_1=30cm{R}_2=10cm$]

Thus Ratio ,

$\frac{{\varphi }_1}{{\varphi }_2}=\frac{R_1^2+R_2^2}{R_1^2-R_2^2}=\frac{{30}^2+{10}^2}{{30}^2-{10}^2}=\frac{1000}{800}=\frac{5}{4}$