Part A: Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf ε = ε₀ cos ωt, three separate wires carry currents for the emfs ε₁ = ε0 cos ωt, ε₂ = ε0 cos(ωt + 120^o), and ε₃ = ε₀ cos(ωt - 120^o). This is why the long- distance transmission lines you see in the countryside have three parallel wires, as do many distribution lines within a city.

Part B: Show that the sum of the three phases is zero, producing what is referred to as neutral. In single-phase electricity, provided by the familiar 120 V/60 Hz electric outlets in your home, one side of the outlet is neutral, as established at a nearby electrical substation. The other, called the hot side, is one of the three phases. (The round opening is connected to ground.)

Part C: Find an expression for the rms value of the potential difference between any two of the phases. Assume that ε_rms is the familiar single-phase rms voltage. Some high-power home appliances, especially electric clothes dryers and hot-water heaters, are designed to operate between two of the phases rather than between one phase and neutral. Heavy-duty industrial motors are designed to operate from all three phases, but full three-phase power is rare in residential or office use.