A current I=2.3 A flows around a continuous path that consists of portions of two concentric circles of radii a and a/2, respectively, where a=6 cm, and two straight radial segments. The point P is at the common center of the two circle segments.
Calculate the components of the magnetic field (in T) at point P.
To calculate the components of the magnetic field at point P, we can use Ampere's law. Ampere's law relates the magnetic field around a closed loop to the electric current passing through the loop.
1. First, let's calculate the magnetic field produced by the current flowing along the concentric circles.
The magnetic field produced by a current flowing along a circle is given by B = μ₀I / (2πr), 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 radius of the circle.
For the current flowing along the circle of radius a, the magnetic field at P is B₁ = μ₀I / (2πa).
For the current flowing along the circle of radius a/2, the magnetic field at P is B₂ = μ₀I / (2π(a/2)).
2. Next, let's calculate the magnetic field produced by the current flowing along the straight radial segments.
The magnetic field produced by a current flowing along a straight wire is given by B = μ₀I / (2πr), 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.
Since the radial segments connect the center of the circles to point P, the distance r is simply the radius a.
For the current flowing along the radial segment from the outer circle to point P, the magnetic field at P is B₃ = μ₀I / (2πa).
For the current flowing along the radial segment from the inner circle to point P, the magnetic field at P is B₄ = μ₀I / (2π(a/2)).
3. Now let's add up the magnetic field components at point P.
B = (B₁ + B₂) - (B₃ + B₄).
Substituting the values, we have:
B = (μ₀I / (2πa)) + (μ₀I / (2π(a/2))) - (μ₀I / (2πa)) - (μ₀I / (2π(a/2))).
Simplifying the expression, we get:
B = μ₀I / (2π(a/2)).
Substituting the value of a, we have:
B = μ₀I / (2π(6/2)).
B = μ₀I / (6π).
Finally, we can calculate the numerical value for B using the given current I and the value of the permeability of free space μ₀.
B = (4π × 10^(-7) T·m/A) * (2.3 A) / (6π).
B ≈ 0.00383 T.
Therefore, the magnetic field at point P is approximately 0.00383 T.