The angular separation between the first-order maxima is given by Δθ = λ/d, where d is the grating spacing. The grating spacing is given by d = 1/5000 m = 2 × 10^(-4) m. Substituting the values, we get Δθ = 500 × 10^(-9) m / 2 × 10^(-4) m = 0.1 rad

The maximum emf induced in the coil is given by ε = N × (dΦ/dt), where N is the number of turns, Φ is the magnetic flux, and d/dt is the derivative with respect to time. The magnetic flux is given by Φ = B × A × cos(ωt), where ω is the angular velocity and t is time. Substituting the values, we get ε = 500 × (0.1 T × 0.02 m^2 × cos(20 rad/s × t))/dt = 6.28 V.

The energy density of the electric field between the plates is given by u = (1/2)ε₀E^2, where ε₀ is the permittivity of free space and E is the electric field strength. The electric field strength is given by E = V/d = 12 V / 2 × 10^(-3) m = 6 × 10^3 V/m. Substituting this value, we get u = (1/2) × 8.85 × 10^(-12) C^2/Nm^2 × (6 × 10^3 V/m)^2 = 1.8 × 10^5 J/m^3.

The force experienced by a charged particle in a magnetic field is given by F = qvB sin(θ), where θ is the angle between the velocity and magnetic field vectors. In this case, since the velocity and magnetic field are perpendicular, sin(θ) = 1, and F = qvB = 3 × 10^(-6) C × 0.6 × 3 × 10^8 m/s × 2 T = 3.6 × 10^(-2) N.

This is because the net magnetic flux through any closed surface is always zero. This is known as Gauss's law for magnetism. It is a consequence of the fact that there are no magnetic monopoles, meaning that magnetic field lines always form closed loops.