Question:

A long straight wire carrying a current of 5.0 A is bent into a circular loop with a radius of 2.0 m. A uniform magnetic field of 0.2 T is applied perpendicular to the plane of the loop. Determine the magnitude of the net magnetic field at the center of the loop.

Answer:

To find the net magnetic field at the center of the loop, we need to consider the contributions from both the magnetic field due to the wire carrying current and the applied uniform magnetic field.

Let's start by calculating the magnetic field at the center of the loop due to the wire carrying current using Ampere's law. For a straight wire, the magnetic field at a distance r from the wire is given by:

B = (μ₀ * I) / (2π * r)

where B is the magnetic field, μ₀ is the permeability of free space (4π × 10^(-7) T m/A), I is the current, and r is the distance from the wire.

At the center of the loop, the distance from the wire is equal to the radius of the loop (r = 2.0 m). Plugging the given values into the formula, we get:

B_wire = (4π × 10^(-7) T m/A * 5.0 A) / (2π * 2.0 m) = (20π × 10^(-7) T m) / (4π m) = 5 × 10^(-7) T

Next, let's calculate the magnetic field at the center of the loop due to the applied uniform magnetic field using the formula:

B_applied = 0.2 T

Now, to find the net magnetic field at the center of the loop, we need to consider the vector sum of B_wire and B_applied. Since the magnetic fields are perpendicular to each other at the center of the loop, the net magnetic field can be found using the Pythagorean theorem:

B_net = √(B_wire^2 + B_applied^2) = √((5 × 10^(-7) T)^2 + (0.2 T)^2) ≈ 0.2000000007 T

Therefore, the magnitude of the net magnetic field at the center of the loop is approximately 0.2 T.

(Note: The tiny discrepancy in the answer is due to rounding errors during calculations and can be ignored in practice.)