Answer
$\theta = 2.00~rad$
Work Step by Step
We can find an expression for the magnetic field at the center due to each semi-infinite straight section:
$B = \frac{\mu_0~i}{4~\pi~R}$
Then the magnetic field at the center due to both semi-infinite straight sections is:
$B = \frac{\mu_0~i}{2~\pi~R}$
By the right hand rule, this magnetic field is directed upward.
We can find an expression for the magnetic field at the center due to the arc of current:
$B = \frac{\mu_0~i~\theta}{4~\pi~R}$
By the right hand rule, this magnetic field is directed downward.
For the magnetic field at the center to be zero, the magnetic field due to the arc of current must be equal in magnitude and opposite in direction to the magnetic field due to the semi-infinite straight sections.
To find $\theta$, we can equate the magnitudes of the magnetic field directed downward and the magnetic field directed upward:
$\frac{\mu_0~i~\theta}{4~\pi~R} = \frac{\mu_0~i}{2~\pi~R}$
$\frac{\theta}{4} = \frac{1}{2}$
$\theta = 2.00~rad$
