Question

Two straight and long conductors AOB and COD are perpendicular to each other and carry currents of $I_{1}and{I}_2$. The magnitude of the magnetic field at a point P at a distance a from point 0 in a direction perpendicular to the plane ABCD is

• A. $\frac{{\mu }_0}{2\pi a}(I_1+I_2)$
• B. $\frac{{\mu }_0}{2\pi a}(I_1-I_2)$
• C. $\frac{{\mu }_0}{2\pi a}(I_1^2+I_2^2{)}^{\frac{1}{2}}$
• D. $\frac{{\mu }_0}{2\pi a}\frac{I_1{I}_2}{(I_1+I_2)}$

Answer 1

The correct option is C

$\frac{{\mu }_0}{2\pi a}(I_1^2+I_2^2{)}^{\frac{1}{2}}$

The magnetic field at point P due to current $I_1$ in conductor AOB is
$B_1=\frac{{\mu }_0{I}_1}{2\pi a}$
and the magnetic field at point P due to current $I_2$ in conductor COD is
$B_2=\frac{{\mu }_0{I}_2}{2\pi a}$
Since the two conductors are perpendicular to each other, fields $B_{1}and{B}_2$ will be perpendicular to each other. Therefore, the resultant field at P is
$B=(B_1^2+B_2^2{)}^{\frac{1}{2}}=\frac{{\mu }_0}{2\pi a}(I_1^2+I_2^2{)}^{\frac{1}{2}}$
Hence the correct choice is (c).