A very thin, infinitely long metal sheet lies in the xy-plane,
between x = -w and x = w. A current of density h A/m flows
in the +y-direction. What are the magnitude and direction
of the magnetic field at a distance z « w above and
below the sheet? Neglect end effects.
I think B = (mu)h
where mu is the permeability of free space. Direction is +x above and -x below.
To determine the magnitude and direction of the magnetic field at a distance z << w above and below the infinitely long metal sheet, you can use the Biot-Savart Law. This law allows you to calculate the magnetic field created by a current-carrying wire or a current-carrying sheet.
The Biot-Savart Law states that the magnetic field dB at a point P due to a small segment of current-carrying wire or sheet dI is given by the formula:
dB = (μ0 / 4π) * (dI x r) / r^3
where:
- dB is the magnetic field created by the current segment dI at point P,
- μ0 is the permeability of free space (approximately 4π x 10^-7 T·m/A),
- dI is the current element,
- r is the position vector from the current segment to the point P,
- x represents the cross product.
For an infinitely long current-carrying sheet with current density h A/m flowing in the +y-direction, we can consider an elemental current strip of width Δx, located at a distance x from the origin.
The length of this elemental current strip is ΔL, which is equal to Δx.
The current I through this strip is given by I = h ΔL.
Now, let's consider a point P that is distance z above the center of the sheet. The elemental current strip lies along the x-axis on the xy-plane, between x = -w and x = w.
The position vector r from the elemental current strip to the point P is r = (x, z).
To find the total magnetic field at point P, we need to integrate the contribution from all the elemental current strips. However, since end effects are neglected, we can assume the contributions from the left and right sides of the sheet cancel each other out, resulting in a net magnetic field only due to the y-directed current density h.
The magnitude of the magnetic field B at point P is given by the integral:
B = ∫ dB
Substituting the values into the Biot-Savart Law equation, we have:
B = (∫ (μ0 / 4π) * (I Δx x r) / r^3) * h
Now we can simplify this equation. Since we are integrating the current density h along x, we can take the h outside of the integral:
B = h * (∫ (μ0 / 4π) * (I Δx x r) / r^3)
Taking all the constants out of the integral, we have:
B = (μ0 * h / (4π)) * (∫ (I Δx x r) / r^3)
Now, we need to evaluate the integral. Note that x will go from -w to w, and Δx will cancel out.
B = (μ0 * h / (4π)) * (∫ (I x r) / r^3) from x = -w to x = w
To calculate the integral, we need to express the position vector r = (x, z) in terms of x. Since we are considering a point P that is z above the center of the sheet, the z component is fixed at z. Hence, r = (x, z).
Now, we insert the limits for x and evaluate the integral:
B = (μ0 * h / (4π)) * ((I * [(w, z)] - I * [(-w, z)]) / [(w^2 + z^2)^(3/2)])
Simplifying further, we have:
B = (μ0 * h / (4π)) * (2I * [(w, z)] / [(w^2 + z^2)^(3/2)])
Since the direction of current density h is in the +y-direction and the position vector r points in the +x direction, using the right-hand rule, we can determine that the magnetic field will circulate counterclockwise around the current sheet.