Final answer:
Water will heat up faster using an immersion heater with a long and thin coil because it generates more heat due to greater resistance and provides better heat transfer with its larger surface area.
Step-by-step explanation:
The water will heat up faster using an immersion heater with a long and thin coil of wire. This is because the longer and thinner wire has a greater resistance, which leads to more heat generation when current is passed through it. According to the electrical power formula (P = I2R where P is power, I is current, and R is resistance), increasing resistance while keeping the current constant results in more power. Moreover, the greater surface area of a long and thin coil provides more contact with the water, facilitating better heat transfer.
In a 120 V circuit, an immersion heater that can boil water in a given time has a specific resistance calculated based on energy, time, and water's specific heat capacity. Reducing the resistance could potentially speed up the heating process; however, this is ultimately limited by the power capacity of the circuit and the safety concerns of adequately dissipating the heat generated.
The transfer of heat from the heater to the water is an application of Joule's law, which states that the heat produced by an electric current is proportional to the resistance of the material, the square of the current, and the time the current is applied. The larger surface area and resistance would result in a faster heating time for the water as more heat is released due to the current passing through the wire.