WBJEE 2026 Physics and Chemistry Paper

Concept:A time-varying magnetic field induces a tangential electric field. The emf across the rod is the line integral of this induced electric field along the rod.Explanation:The magnetic field increases at a constant rate, so $\frac{dB}{dt} = \alpha$.For a point at distance $r$ from the centre, the induced electric field is $E = \frac{r\alpha}{2}$ for $r \le R$ and $E = \frac{R^2\alpha}{2r}$ for $r \ge R$.The rod is tangent to the circular region, with its midpoint at the tangent point $(0, R)$ and its endpoints at $(-R, R)$ and $(R, R)$.At any point $(x, R)$ on the rod, the distance from the centre is $r = \sqrt{x^2 + R^2}$, so $E = \frac{R^2\alpha}{2\sqrt{x^2 + R^2}}$.The tangential field has a component along the rod: $E_x = -\frac{R^3\alpha}{2(x^2 + R^2)}$.Hence, $\varepsilon = \int_{-R}^{R} E_x \, dx = -\frac{R^2\alpha}{2}\left[\tan^{-1}\frac{x}{R}\right]_{-R}^{R} = -\frac{\pi R^2\alpha}{4}$.Taking the magnitude, the induced emf is $|\varepsilon| = \frac{1}{4}\pi R^2\alpha$.Answer:$\frac{1}{4}\pi R^2\alpha$, i.e., Option D.