When a good conductor is exposed to an oscillating electromagnetic field, currents flow near the surface and dissipate energy as heat. These currents oppose the changing field, so the wave loses amplitude as it penetrates the material. The depth at which the wave amplitude falls to 1/e of its surface value is called the skin depth, δ. In a good conductor, solving Maxwell’s equations for a plane wave gives a complex propagation constant whose magnitude leads to α ≈ β ≈ sqrt(ω μ σ / 2), where α is the attenuation constant. The skin depth is the reciprocal of that attenuation, so δ = 1/α = sqrt(2/(μ σ ω)). This shows that higher conductivity, higher frequency, or greater permeability shrink the skin depth and cause stronger attenuation of the wave as it goes in. That is why a good conductor causes strong attenuation of the wave. The other formulas either miss the correct factors or describe other behaviors (like amplification or complete reflection) that don’t match how skin depth depends on μ, σ, and ω.

The essential idea is to use the Biot–Savart law and integrate along the finite straight wire. Each little segment dl carries current I and contributes a small magnetic field that, when you sum all along the wire, adds up to a simple result at a point a distance d away from the wire. Because the observation point is at a fixed perpendicular distance d, the geometry depends only on how the lines from the observation point to the wire’s ends subtend angles with the wire’s axis. Denoting those end angles by θ1 and θ2, the integration over the wire gives a magnitude B = μ0 I /(4π d) times (cos θ1 − cos θ2). The cos terms come from the way the contributions add up with the varying direction along the wire, and the difference reflects the finite length. This expression reduces to the well-known infinite-wire result in the appropriate limit: as the ends recede to infinity, the angles arrange so that (cos θ1 − cos θ2) becomes 2, giving B = μ0 I /(2π d), the field of an infinite straight wire. The direction of B is tangential around the wire, given by the right-hand rule.

The key idea is how currents relate to turns in an ideal transformer. Power is conserved, so V1 I1 = V2 I2, and voltages scale with turns: V1/V2 = N1/N2. With a turns ratio of 2:1, the primary has twice as many turns as the secondary, which means V1 is twice V2. To keep power equal, the primary current must be half of the secondary current: I1 = I2 × (N2/N1) = 3 A × (1/2) = 1.5 A. So the primary current is 1.5 A.

An induced electromotive force appears when the magnetic flux through a loop changes in time, and its direction follows Lenz’s law to oppose that change. The magnetic flux through the loop is Φ_B = ∫ B · dA, and the integral form of Faraday’s law is expressed as the line integral of the electric field around the loop: ∇ E · dl = - dΦ_B/dt. The EMF around the loop, often denoted ε, is this line integral, so ε = - dΦ_B/dt. This captures both that the emf is driven by how quickly the flux is changing and that the change is opposed by the induced current. The other expressions fail to match this relation because they miss the minus sign, representing the opposing direction required by Lenz’s law; or they use a simple division by dt or a finite change ΔΦ_B/dt instead of the instantaneous rate of change dΦ_B/dt. The correct form explicitly ties the emf to the instantaneous rate at which the magnetic flux through the loop is changing, with the sign enforcing opposition to that change.