Final answer:
To calculate the length of the resistor of a circular wire used in a heater element, one must first calculate the resistance using the power formula and then rearrange the resistance formula to solve for length considering the wire's cross-sectional area and the resistivity of aluminium.
Step-by-step explanation:
To calculate the length of the resistor of a circular wire used in a heater element connected to a 240V supply where the heater consumes 8 joules of energy in 0.25 seconds, we use the formula for power P = V2 / R, where P is power, V is voltage, and R is resistance. To find R, we can rearrange the formula to R = V2 / P. Since power is also the energy consumed per unit time (E/t), we find that P = 8 J / 0.25 s = 32 watts. Thus, R = (240V)2 / 32W ≈ 1800 Ω.
To find the length of the wire (L), we use the formula R = ρ * (L/A), where ρ is the resistivity of the material and A is the cross-sectional area. The resistivity of aluminium is given as 2.825 x 10⁻⁶ Ωcm. The cross-sectional area A is π * (d / 2)2, with d being the diameter of the wire. We convert the diameter from inches to meters: 0.00016 inches * 25.4 mm/inch * 0.1cm/mm * 0.01m/cm = 0.00004064 m. So, A = π * (0.00004064 m / 2)2.
Finally, we calculate L by rearranging the formula for resistance: L = R * A / ρ. With all values in correct units, we can solve for the length of the resistor wire in meters.