Consider two N-turn circular coils of radius R, each perpendicular to the axis of symmetry, with their centers located at z=±l/2 . There is a steady current I flowing in the same direction around each coil
Assuming the N=100, I=2 Amperes, R=10 cm, l=2 cm calculate the magnitude of the magnetic fied (in Tesla) at a distance z=0.5 cm from the midpoint between the centers of the two coil.
Calculate the second derivative with respect to z of the magnetic field calculated in part (b) using the same values for the parameters. Express your answer in Tesla per meter^2.
To calculate the magnitude of the magnetic field at a given point, we can use the Biot-Savart law. The Biot-Savart law relates the magnetic field at a point due to a current-carrying wire segment.
The equation for the magnetic field at a point P, located at a distance r from a short wire segment of length dl carrying current I, is given by:
dB = (μ₀ * I * dl * sinθ) / (4π * r²),
where μ₀ is the permeability of free space, I is the current, dl is the differential length element of the wire, θ is the angle between the wire segment and the line connecting the wire to the point P, and r is the distance between the wire and the point P.
In this case, we have two circular coils with N turns carrying a steady current I in the same direction. The radius of each coil is R, and their centers are located at z = ±l/2. We want to calculate the magnetic field at a point located at z = 0.5 cm (0.005 m) from the midpoint between the coil centers.
To calculate the magnetic field due to each coil, we need to integrate the magnetic field contributions from each turn of the coil. Since the coils are symmetric and perpendicular to the z-axis, the angle θ will be constant for each turn in the formula.
Let's denote the angle between the line connecting the midpoint between the coil centers and the current element dl as α. Then α can be calculated as:
α = arctan(l/2) = arctan(0.02).
Now we can calculate the magnetic field contribution dB from each coil at the given point:
dB = (μ₀ * I * dl * sinα) / (4π * r²).
To find the total magnetic field B at the given point, we sum the contributions from each coil. Since the coils are symmetric and carrying current in the same direction, the net contribution of the magnetic field from the two coils will be twice the magnetic field from a single coil.
B = 2 * ∫dB.
Now we can substitute the values into the formula:
μ₀ = 4π * 10^(-7) T·m/A (permeability of free space)
I = 2 A (current)
l = 0.02 m (distance between the coil centers)
r = 0.005 m (distance from the midpoint between the coil centers)
N = 100 (number of turns per coil)
First, let's calculate the magnetic field dB from a single coil at the given point:
dB = (μ₀ * I * dl * sinα) / (4π * r²)
= (4π * 10^(-7) * 2 * dl * sin(arctan(0.02))) / (4π * 0.005²).
We need to integrate this expression over the length of the coil, which is given by 2πR.
∫dB = ∫(4π * 10^(-7) * 2 * dl * sin(arctan(0.02))) / (4π * 0.005²) from 0 to 2πR.
Simplifying and integrating:
∫dB = (4π * 10^(-7) * 2 * sin(arctan(0.02))) / (4π * 0.005²) * ∫dl from 0 to 2πR
= (4 * 10^(-7) * sin(arctan(0.02))) / (0.005²) * 2πR.
Now, substituting the values:
∫dB = (4 * 10^(-7) * sin(arctan(0.02))) / (0.005²) * 2π * 10 * 0.02
= 8π * 10^(-7) * sin(arctan(0.02)).
Since we have two coils, the total magnetic field at the given point is:
B = 2 * ∫dB
= 2 * 8π * 10^(-7) * sin(arctan(0.02)).
Now we can substitute the values into the formula:
B = 2 * 8π * 10^(-7) * sin(arctan(0.02)).
Calculating this expression will give us the magnitude of the magnetic field at the given point.