Final answer:
To run the power supply at maximum power, the electromagnet coil would require approximately 842 turns, calculated by determining the current, resistance of the copper wire, and then using the resistivity of copper to estimate the wire length and number of turns.
Step-by-step explanation:
To calculate the number of turns needed for the coil to run the power supply at maximum power, we must first understand the relationship between voltage (V), current (I), power (P), and resistance (R). According to Ohm's Law and the power formula:
P = IV
Since P is given as 1.0 kW and V as 34 V, we can find the current I:
I = P / V = 1000 W / 34 V = 29.41 A (approximately)
The resistance of the coil can then be calculated using Ohm's Law:
R = V / I = 34 V / 29.41 A = 1.16 Ω (approximately)
Now, we need to use the resistivity formula for a wire, which is:
R = ρ(L/A)
Where ρ is the resistivity of copper (approximately 1.68×10⁻⁸ Ω·m), L is the length of the wire, and A is the cross-sectional area. The wire is square with each side being 2.4 mm, so:
A = (2.4×10⁻³ m) ² = 5.76×10⁻⁶ m²
The circumference of the coil is πd, where d is the diameter of the coil:
C = πd = 3.14 * 1.5 m = 4.71 m
Now let's use the length of one turn (which is the circumference) to find the total length of wire needed for the resistance we calculated:
L = R * (A / ρ) = 1.16 Ω * (5.76×10⁻⁶ m² / 1.68×10⁻⁸ Ω·m) = 3968.45 m (approximately)
Finally, we calculate the number of turns N by dividing the total length L by the circumference C:
N = L / C = 3968.45 m / 4.71 m ≈ 842 turns (approximately)
To summarize, approximately 842 turns are needed to run the power supply at maximum power.