an insulated circular ring of diameter 6.50cm carries a 12.0 A current and is tangent to a very long, straight insulated wire carrying 10.0 A of current. find the magnitude and direction of the magnetic field at the center of the ring due to this combination of wires.

To find the magnitude and direction of the magnetic field at the center of the ring due to the combination of wires, you can follow these steps:

Step 1: Calculate the magnetic field at the center of the ring due to the straight wire.

Use Ampere's Law to calculate the magnetic field at a distance r from a long straight wire carrying current I:

B = (μ₀ * I) / (2π * r),

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

In this case, the distance from the straight wire to the center of the ring is the radius of the ring divided by 2 (since the ring is tangent to the wire):

r₁ = 6.50 cm / 2 = 3.25 cm = 0.0325 m.

Substituting the values into the formula, we get:

B₁ = (4π × 10⁻⁷ T·m/A * 10.0 A) / (2π * 0.0325 m).

Calculating B₁:

B₁ = (4π × 10⁻⁷ T·m/A * 10.0 A) / (2π * 0.0325 m).

B₁ ≈ 3.08 × 10⁻⁵ T.

Step 2: Calculate the magnetic field at the center of the ring due to the circular wire.

The magnetic field at the center of a circular loop of current is given by:

B = (μ₀ * I) / (2R),

where B is the magnetic field, μ₀ is the permeability of free space, I is the current, and R is the radius of the loop.

In this case, the radius of the loop is half of the diameter:

R = 6.50 cm / 2 = 3.25 cm = 0.0325 m.

Substituting the values into the formula, we get:

B₂ = (4π × 10⁻⁷ T·m/A * 12.0 A) / (2 * 0.0325 m).

Calculating B₂:

B₂ = (4π × 10⁻⁷ T·m/A * 12.0 A) / (2 * 0.0325 m).

B₂ ≈ 3.70 × 10⁻⁴ T.

Step 3: Calculate the net magnetic field at the center of the ring.

Since the magnetic fields due to both the straight wire and the circular loop are along the same direction (tangent to the ring), you can add them algebraically to get the net magnetic field:

B_net = B₁ + B₂.

Substituting the values:

B_net ≈ 3.08 × 10⁻⁵ T + 3.70 × 10⁻⁴ T.

Calculating B_net:

B_net ≈ 4.01 × 10⁻⁴ T.

So, the magnitude of the magnetic field at the center of the ring due to the combination of wires is approximately 4.01 × 10⁻⁴ Tesla.

To determine the direction, you can use the right-hand rule. Place your right hand in the direction of the current in the straight wire. Your thumb will point in the direction of the magnetic field at the center of the ring.