Circular loop of a wire and a long straight wire carry currents I_c...

Magnetic field due to a circular loop at the centre of the loop is $B_1=\frac{\mu_0}{4 \pi} \frac{2 \pi I_c}{R}$. Magnetic field due to a straight line at the centre of the loop is $B_2=\frac{\mu_0 2 I_e}{4 \pi H}$ Both these fields act in opposite direction hence total magnetic field at the centre of the loop is $$\begin{aligned}& B=B_1-B_2=\frac{\mu_0 2 \pi I_c}{4 \pi R}-\frac{\mu_0 2 I_e}{4 \pi H}\\ & B=0 \text{ (given). }\\ & \therefore \frac{\mu_0 2 \pi I_c}{4 \pi R}=\frac{\mu_0 2 I_e}{4 \pi H} \text{ or, }H=\frac{I_e R}{\pi I_c} . \end{aligned}$$