A current flows through a conductor arranged in one plane as shown in figure. The equation of ellipse guiding the diversion of current in the conductor is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Magnetic field due to given current distribution at O is given by ${\mu }_{0}i\left\{\frac{n}{a}+\frac{m}{b}\right\}(wherem,narenon-negative)$. Then value of m + n is:

The element of current of conductor ACB and ADB symmetrical to each other induces same magnetic field, but directed oppositely. Hence, the field at the arbitrary point on AB induced by the entire conductor is zero, since the straight sections of the conductor do not induce a field on AB.

 $\begin{array}{l}{B}_{net}={\mu }_0 i \left(\frac{m}{a}+\frac{n}{b}\right)=0 \left(given\right)\\ \text{}here a and b are positive numbers.\\ \text{It means to satisfy above given condition, either of m or n must be negative or both are zero.}\\ Since, m \& n, are non-negative,\\ \Rightarrow m=n=0\text{}\\ \Rightarrow m+n=0\end{array}$