Magnetic Field Due to a Circular Loop

To calculate the magnetic field due to a circular loop at a point P, we can use Ampere's law. Ampere's law states that the magnetic field at a point due to a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire.

Given:
- Length of the wire (circumference of the loop) = 0.26 m
- Current flowing through the loop = 2A
- Distance from the center of the loop to point P = 0.15 m

Step 1: Calculate the Radius of the Circular Loop
The length of the wire is equal to the circumference of the loop, which is given by the formula:
C = 2πr, where C is the circumference and r is the radius of the loop.

Given C = 0.26 m, we can rearrange the formula to solve for r:
r = C / (2π)
r = 0.26 m / (2π)
r ≈ 0.0414 m

So, the radius of the circular loop is approximately 0.0414 m.

Step 2: Calculate the Magnetic Field at Point P
Using Ampere's law, we can calculate the magnetic field at point P, which is at a distance of 0.15 m from the center of the loop.

The formula for the magnetic field due to a circular loop at a point on its axis is given by:
B = (μ₀ * I * R²) / (2 * (R² + x²)^(3/2))

Where:
B is the magnetic field at point P,
μ₀ is the permeability of free space (constant),
I is the current flowing through the loop,
R is the radius of the loop,
x is the distance from the center of the loop to point P.

Using the given values:
μ₀ = 4π × 10^(-7) T·m/A (permeability of free space)
I = 2 A (current flowing through the loop)
R = 0.0414 m (radius of the loop)
x = 0.15 m (distance from the center of the loop to point P)

Plugging in these values into the formula, we can calculate the magnetic field at point P.

Step 3: Calculation
B = (4π × 10^(-7) T·m/A * 2 A * (0.0414 m)²) / (2 * ((0.0414 m)² + (0.15 m)²)^(3/2))

Simplifying the equation, we get:
B ≈ 1.6 × 10^(-6) T

Therefore, the magnetic field due to the circular loop at point P is approximately 1.6 × 10^(-6) T.