You know you should not have bought the "MRI in a Box" kit from the Home Shopping Channel. Its main solenoid is supposed to have a radius of r=0.28 m, a length of L=0.1 m, and 12000 windings. The wire they give you has a resistance of 13 mΩ per meter. With only air inside, it's supposed to generate a field of 6.3 Tesla.
A.) What current would be needed?
B.) How much power would your MRI device consume in steady operation?
So I found the current of the solenoid to be 41.8 A.
But for part B, I couldn't figure out how much power is consumed. I thought I could use P = I^2 * R and use the resistance of 13 mOhms per meter (P = 41.8 A ^2 * 0.0013 ohms) but I still come out with the wrong answer.
To find the power consumed by the MRI device in steady operation, you need to consider the entire length of the wire used in the solenoid.
Let's break down the solution step by step:
A.) Finding the current needed:
The magnetic field inside a solenoid can be calculated using the formula: B = μ₀ * n * I, where B is the magnetic field, μ₀ is the permeability of free space (equal to 4π * 10^-7 Tm/A), n is the number of turns per unit length (windings/meter), and I is the current.
Given that the magnetic field (B) is 6.3 Tesla, the number of windings (n) is 12000 windings, and the radius (r) and length (L) of the solenoid are 0.28 m and 0.1 m, respectively, we can determine the number of turns per unit length:
n = total windings / total length = 12000 / 0.1 m = 120000 windings/m
Now we can solve for the current (I):
B = μ₀ * n * I
6.3 T = (4π * 10^-7 Tm/A) * (120000 windings/m) * I
Solving for I:
I = (6.3 T) / [(4π * 10^-7 Tm/A) * (120000 windings/m)]
I ≈ 41.83 A
So the current needed for the solenoid is approximately 41.83 A.
B.) Finding the power consumed:
To calculate the power consumed (P), you need to consider the resistance (R) of the wire used in the solenoid.
The resistance of the wire is given as 13 mΩ per meter. Since the solenoid has a length of 0.1 m, the total resistance (R) of the solenoid can be calculated as:
R = resistance per meter * length = 0.013 Ω/m * 0.1 m = 0.0013 Ω
However, the power formula P = I^2 * R should be modified because the resistance of the wire is not constant along its length due to the varying magnetic field inside the solenoid. The magnetic field causes a non-uniform distribution of current, resulting in a varying resistance.
To accurately calculate the power consumed, we need to integrate the power formula over the entire length of the wire. However, this integration is beyond the scope of a simple explanation.
Thus, it seems that the information provided in the problem may not be sufficient to accurately determine the power consumed by the MRI device in steady operation using the given wire resistance.
It's worth noting that in real MRI devices, the power consumed by the solenoid is typically much higher due to factors such as energy losses in the coils, eddy currents, and heat dissipation.