The power required to produce a magnetic field of 8.00 mT at the center of the solenoid is approximately 1.19 × 10¹⁹ W.
To calculate the power required to produce a magnetic field of 8.00 mT at the center of the solenoid, we can use the following formula:
P = I² × R
where:
P is the power in watts (W)
I is the current in amperes (A)
R is the resistance in ohms (Ω)
The current required to produce the desired magnetic field can be calculated using Ampère's law:
B = μ₀ × n × I
where:
B is the magnetic field in teslas (T)
μ₀ is the permeability of free space (4π × 10⁻⁷ T⋅m/A)
n is the number of turns per meter (turns/m)
I is the current in amperes (A)
The resistance of the solenoid can be calculated using the following formula:
R = ρ × L / A
where:
R is the resistance in ohms (Ω)
ρ is the resistivity of copper (1.68 × 10⁻⁸ Ω·m)
L is the length of the solenoid in meters (m)
A is the cross-sectional area of the wire in square meters (m²)
Plugging in the values, we get:
P = (8.00 × 10⁻³ T)² × (4π × 10⁻⁷ T·m/A)² × (750 turns / 0.75 m) / (1.68 × 10⁻⁸ Ω·m) × (π × (0.05 m)²)) ≈ 1.19 × 10¹⁹ W