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 Bz of the magnetic field (in T) at point P.
2e-03
1e-03
0
how is you process?
To calculate the components Bz of the magnetic field at point P, we can use the Biot-Savart Law. The law states that the magnetic field at a point due to a current-carrying wire or a current loop can be calculated by summing the contributions from each infinitesimal element of the wire or loop.
In this case, we have a continuous path consisting of portions of two concentric circles and two straight radial segments. Let's break down the calculation into separate components and calculate them individually.
1. Contribution from the concentric circles:
The magnetic field at a point on the axis of a circular current loop is given by the formula:
Bz = (μ0 I a^2)/(2(x^2 + a^2)^(3/2))
Considering the circle with radius a (outer circle), the distance between the center of the circle and point P is x = a. Plugging the values into the formula, we get:
Bz1 = (μ0 I a^2)/(2(a^2 + a^2)^(3/2))
= (μ0 I a^2)/(2(2a^2)^(3/2))
= (μ0 I a^2)/(2 * 2^(3/2) * a^3)
= (μ0 I)/(4 * √2 * a)
2. Contribution from the smaller concentric circle:
Similarly, considering the circle with radius a/2 (inner circle), the distance between the center of the circle and point P is x = a/2. Plugging the values into the formula, we get:
Bz2 = (μ0 I (a/2)^2)/(2((a/2)^2 + (a/2)^2)^(3/2))
= (μ0 I (a/2)^2)/(2 * (2(a/2)^2)^(3/2))
= (μ0 I (a/2)^2)/(2 * 2^(3/2) * (a/2)^3)
= (μ0 I)/(16 * √2 * a)
3. Contribution from the straight radial segments:
The magnetic field at a point on the axis of a straight current-carrying wire is given by the formula:
Bz = (μ0 I)/(2πx)
Considering each straight radial segment, the distance between the point P and the wire is equal to the radius of the corresponding circle (a or a/2). Let's denote the length of the straight segments as L. The total contribution from the two straight radial segments is given by:
Bz3 = (μ0 I)/(2πa) + (μ0 I)/(2π(a/2))
= (μ0 I)/(2π) * (1/a + 2/a)
= (μ0 I)/(2πa) * (3/2)
= (3μ0 I)/(4πa)
Now, to find the total magnetic field at point P, we need to add up the contributions from each component:
Bz_total = Bz1 + Bz2 + Bz3
= (μ0 I)/(4 * √2 * a) + (μ0 I)/(16 * √2 * a) + (3μ0 I)/(4πa)
= (μ0 I * (4 + 1))/(16 * √2 * a) + (3μ0 I)/(4πa)
= (5μ0 I + 12πμ0 I)/(16 * √2 * a * 4πa)
= (17μ0 I)/(256πa^2√2)
Therefore, the component Bz of the magnetic field at point P is (17μ0 I)/(256πa^2√2) (in tesla).