A toroid is fabricated with a circular shape and loops with a square cross section as shown in Fig. 28–69. The cross section of a loop is a square of side 6.0 cm. The inner radius of the whole circular toroid is 3.0 m. There are 320 loops of wire which carry a 45-A dc current using a nearby power supply at 20.0 V. The arrows show the current flow in and out of the toroid. The current flows up at the inner diameter and down at the outer diameter. (a) Calculate the strength of the magnetic field at the center of the square’s cross section at point P. (b) Is the magnetic field pointing clockwise or counterclockwise? (c) The square cross sectional area of the wire is uniformly . What is the resistivity of the wire?

<IMAGE>

Verified step by step guidance

1. Step 1: Use Ampere's Law to calculate the magnetic field inside the toroid. Ampere's Law states that the integral of the magnetic field (B) along a closed loop is equal to the permeability of free space (μ₀) times the enclosed current (I_enc). For the toroid, the path of integration is a circle at the radius of interest, and the enclosed current is the total current through all the loops.
2. Step 2: Calculate the enclosed current (I_enc) by multiplying the current per loop by the number of loops. In this case, each loop carries a current of 45 A and there are 320 loops, so I_enc = 45 A * 320.
3. Step 3: Apply Ampere's Law specifically for the toroid. The magnetic field (B) around the toroid is given by B = (μ₀ * I_enc) / (2πr), where r is the radius at which you want to find the magnetic field. Here, you would use the average radius of the toroid, which is the average of the inner and outer radii.
4. Step 4: Determine the direction of the magnetic field using the right-hand rule. Point your thumb in the direction of the current and curl your fingers. Your fingers will curl in the direction of the magnetic field around the wire.
5. Step 5: Calculate the resistivity of the wire using Ohm's Law and the given dimensions of the wire. Ohm's Law states V = IR, where V is the voltage, I is the current, and R is the resistance. The resistance can be further expressed in terms of resistivity (ρ), length (L), and cross-sectional area (A) as R = ρL/A. Use the given voltage, current, and dimensions of the wire to find ρ.