Question

Two identical coils carry equal currents have a common center and their planes are at right angles to each other. The ratio of the magnitude of the resultant magnetic field at the center and the field due to one coil is

• A. 2 :1
• B. 1 :2
• C. $\sqrt{2}:1$
• D. $1:\sqrt{2}$

Answer 1

The correct option is C $\sqrt{2}:1$

To find: rato of resultant magnetic field to magnetic field of one coil$=?$

Solution: Let radius of both coils be R.

As we know that,

magnetic field at centre due to circular coil is,

$B=\frac{{\mu }_{0}I}{2r}$

Now, $B_1:$

$B_1=\frac{{\mu }_{0}I}{2R}=B\left(say\right)$

Now, $B_2:$

$B_2=\frac{{\mu }_{0}I}{2R}$

$==>B_2=B_1=B$

$B_{eq}=\sqrt{B_1^2+B_2^2}$

$B_{eq}=\sqrt{B^2+B^2}$

$B_{eq}=\sqrt{2B^2}$

$B_{eq}=\sqrt{2}B$

Now,

$\frac{B_{eq}}{B_1}=\frac{\sqrt{2}B}{B}$

$B_{eq}:B_1=\sqrt{2}:1$

as $B_1=B_2$

therefore,$B_{eq}:B_2=\sqrt{2}:1$

hence,

The correct opt : C