A uniformly charged semicircular arc of radius '$$r$$' has linear charge density $$(\lambda)$$, is the electric field at its centre? ( $$\in_0=$$ permittivity of free space)

The P.E. 'U' of a moving particle of mass 'm' varies with 'x'-axis as shown in figure. The deBroglie wavelength or the particle in the regions $$0 \leq x \leq 1$$ and $$x > 1$$ are $$\lambda_1$$ and $$\lambda_2$$ respectively. II the total energy of the particle is '$$\mathrm{nE}$$', then the ratio $$\lambda_1 / \lambda_2$$ is

An inductive coil has a resistance of $$100 ~\Omega$$. When an a.c. signal of frequency $$1000 \mathrm{~Hz}$$ is applied to the coil the voltage leads the current by $$45^{\circ}$$. The inductance of the coil is $$\left(\tan 45^{\circ}=1\right.$$)

The ratio of radii of gyration of a circular ring and circular disc of the same mass and radius, about an axis passing through their centres and perpendicular to their planes is