Q.A uniform magnetic field B(t), pointing upward, fills a circular region of radius s in a horizontal plane. If B is changing with time, find the induced electric field.

[!TLDR] Apply $\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi}{dt}$ around a circle of radius s at the edge of the field region: $E(2\pi s)=-\pi s^2\fr …

A time-varying magnetic field generates an induced electric field, governed by the integral form of Faraday's law, $\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}$. By the symmetry of the problem (B uniform and vertical throughout a circular region of radius s), the induced E field circulates in concentric circles centred on the region's axis, with constant magnitude at a given radius. Applying the loop integral around a circle of radius s (the EDGE of the field region, where the full flux $\Phi=\pi s^2B$ is enclosed): $$E(2\pi s) …

Apply the integral form of Faraday's law, $\oint\vec{E}\cdot d\vec{l}=-d\Phi/dt$, around a circular path of radius s at the EDGE of the field re …

Forgetting the factor of $\frac{1}{2}$ that appears after dividing by the circumference $2\pi s$, and writin …