A uniform but time-varying magnetic field $B(t)$ exists in a circular region of radius $a$ and is directed into the plane of the paper, as shown. The magnitude of the induced electric field at point $P$ at a distance $r$ from the centre of the circular region 

Using Faraday's law for a circular path of radius $r$ centered on the field region, $$\oint \vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}.$$ From the figure, point $P$ is outside the magnetic region, so $r>a$. The magnetic flux linked by this path comes only from the field-filled circle of radius $a$, hence $$\Phi_B=B(t)\pi a^2.$$ Therefore, $$E(2\pi r)=\pi a^2\left|\frac{dB}{dt}\right|.$$ So the magnitude of the induced electric field is $$E=\frac{a^2}{2r}\left|\frac{dB}{dt}\right|.$$ Thus it varies inversely with $r$, so after checking the options, the correct choice is $\text{B}$.