The electric flux through the curved surface is proportional to the charge enclosed, which remains constant, so the flux through the curved surface doubles if the height is doubled.

The electric flux through the curved surface of a cylinder is given by Φ = Q_enc/ε₀, where Q_enc = Q.

The electric field due to a uniformly charged infinite cylinder depends only on the charge per unit length, not on the radius.

Gauss's law applies to electric fields, not magnetic fields. The electric field around a current-carrying wire is not defined by Gauss's law.

The electric field outside the cylindrical surface is directly proportional to the distance from the wire.

Using Gauss's law, the electric field outside the cylinder is E = Q/(2πε₀R).

Using Gauss's law, the electric field E at a distance R from the axis of a long charged cylinder is E = Q/(2πε₀L) for points outside the cylinder.

Stress is defined as force per unit area. Doubling the diameter increases the area by a factor of four, thus reducing the stress.

Tensile stress is given by force/area. Halving the radius reduces the area by a factor of four, thus the stress quadruples for the same force.

Resistance R = ρ(L/A) = 1.68 x 10^-8 * (1 / (π(0.5 x 10^-3)²)) = 0.0424 Ω.

Damping ratio (ζ) = c / (2√(mk)) = 0.5 / (2√(2*8)) = 0.5 / (2√16) = 0.5 / 8 = 0.0625.

Amplitude after time t = A0 * e^(-t/τ) = 10 * e^(-6/3) = 10 * e^(-2) ≈ 2.5 m.

Time constant (τ) = m/c, thus c = m/τ = 1/3 = 0.333 kg/s. Using c = 2ζ√(mk), we find ζ = 0.5.

Critical angle θc = sin^(-1)(1/n) = sin^(-1)(1/2.42) ≈ 24.4°.

Numbers greater than 4 are 5 and 6. Probability = 2/6 = 1/3.

The possible outcomes greater than 2 are {3, 4, 5, 6}. The even numbers among these are {4, 6}. Thus, the probability is 2/4 = 1/2.

Even numbers on a die: 2, 4, 6. Probability = 3/6 = 1/2.

The possible outcomes greater than 2 are {3, 4, 5, 6}. The even numbers among these are {4, 6}. Thus, the probability is 2/3.

The even numbers on a die are 2, 4, and 6, which gives us 3 favorable outcomes. The total outcomes are 6. Therefore, the probability is 3/6 = 1/2.

The dipole moment p = q * d.

The dipole moment p is defined as p = q * d.

Dipole moment p = q * d = (1 × 10^-6 C) * (0.1 m) = 1 × 10^-7 C m.

Dipole moment p = q * d, if d is doubled, p also doubles.

Dipole moment p = q * d = (2 × 10^-6 C) * (0.1 m) = 2 × 10^-7 C m.

Torque τ = p × E = pE sin θ, where θ is the angle between p and E.

Angular momentum L = Iω, where I = (1/2)MR^2 for a disc.

The moment of inertia I of a disc about its axis is (1/2)MR^2. Therefore, the kinetic energy K.E. = (1/2)Iω^2 = (1/2)(1/2)MR^2ω^2 = (1/4)MR^2ω^2.

Kinetic energy K = (1/2)Iω^2, where I = (1/2)MR^2 for a disc.

The relationship between linear speed and angular speed for rolling without slipping is ω = v/R.

The disk has a lower moment of inertia compared to the ring, thus it reaches the ground first.