A metal wire of cross-sectional area 0.5 mm² and length 100 m is connected across a battery of e.m.f. 2 V and internal resistance 1 Ω. The density, atomic mass and electrical conductivity of the metal are 6.35 × 10³ kg m⁻³, 63.5 gm/mole and 2 × 10⁸ mho m⁻¹, respectively. Assuming one conduction electron per atom of the metal, the drift velocity (in mm s⁻¹) of the electrons in the wire is:

[Take Avogadro’s number as 6 × 10²³ and charge of the electron as 1.6 × 10⁻¹⁹ C.]

A nuclear reactor starts producing a radioactive nuclide X from t = 0, at a constant rate of α per second. Each decay of X produces energy E₀, which is utilized to heat a liquid of mass m and specific heat s. Assuming no heat loss from the liquid and taking λ as the decay constant of X, the rate of increase in the temperature of the liquid is :

A beam of polychromatic light passes through a thin prism of prism angle $6^{\circ}$. The refractive index of the material of the prism varies with wavelength $(\lambda)$ as $n(\lambda) = \alpha \lambda + \frac{\beta}{\lambda^2}$, where $\alpha = 3\ \mu m^{-1}$ and $\beta = 0.096\ \mu m^2$. If $\lambda_{min}$ is the wavelength at which the angle of minimum deviation $D_m$ is smallest, then the correct value of $D_m$ at $\lambda_{min}$ is :

A particle of mass m, and angular momentum ℓ is moving in a circular orbit of radius r₀ under the influence of an attractive force $\vec{F}(r)=-\frac{k}{r^2} \hat{r}$. Keeping its angular momentum unchanged, the particle is displaced radially by a small distance $\delta r \ll r_0$, due to which its radial distance varies periodically. The corresponding time period is :