Plane x = 10 carries current 100 mA/m along az while line x = 1, y = —2 carries filamentary current 20π mA along az Determine H at (4, 3, 2).
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To determine the magnetic field intensity (H) at the point (4, 3, 2), we can utilize Ampere's circuital law.
Step 1: Identify the closed path for Ampere's circuital law.
Since the plane x = 10 carries current along az and the line x = 1, y = -2 carries current along az as well, we can choose a rectangular path that includes both these current-carrying elements. Let's define the path as follows:
- Starting point: (1, -2, 2)
- Endpoint: (4, -2, 2)
- Along the path, let's move from (1, -2, 2) to (4, -2, 2), then to (4, 3, 2), and finally back to (1, 3, 2).
Step 2: Enclose the path with an imaginary closed loop.
We enclose the selected path with a rectangle. Let's label the sides of the rectangle as follows:
- AB: Connects (1, -2, 2) and (4, -2, 2)
- BC: Connects (4, -2, 2) and (4, 3, 2)
- CD: Connects (4, 3, 2) and (1, 3, 2)
- DA: Connects (1, 3, 2) and (1, -2, 2)
Step 3: Calculate the line integral of H along the chosen path.
The line integral of H along the chosen path can be expressed as:
∮H⋅dl = H⋅l = μ₀I_enclosed,
where μ₀ is the permeability of free space and I_enclosed is the total enclosed current.
Step 4: Calculate the enclosed current.
The enclosed current is the sum of the currents flowing through the plane and the line within the enclosed area. Given:
- Plane x = 10 carries a current of 100 mA/m in the az direction.
- Line x = 1, y = -2 carries a current of 20π mA in the az direction.
The length of the path AB is 3 units (4 - 1).
Therefore, the current enclosed within the rectangle is:
I_enclosed = Current in the plane + Current in the line
= (100 mA/m) * 3 m + (20π mA) * 3 m
= 300 mA + 60π mA.
Step 5: Calculate H at the given point.
Substituting the values into Ampere's circuital law equation:
H⋅l = μ₀I_enclosed,
we can solve for H:
H = μ₀I_enclosed / l
= [4π × 10^(-7) T*m/A] * [300 mA + 60π mA] / 3 m
Calculating the numerical value of H at the point (4, 3, 2) is left as an exercise as the exact values of μ₀ and π are required.
To determine the magnetic field intensity (H) at the point (4, 3, 2) due to the given current sources, we can use the Biot-Savart law. The Biot-Savart 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.
1. For the plane carrying current, we need to calculate the contribution to the magnetic field from this source. The plane x = 10 is parallel to the yz-plane and carries a current of 100 mA/m along the az direction. Since this plane is infinite, the contribution to the magnetic field at the point (4, 3, 2) will only depend on the y and z coordinates.
2. The contribution to the magnetic field at the point (4, 3, 2) from the plane can be calculated using the Biot-Savart law as follows:
dH_plane = (μ₀/4π) * I_plane * dl_plane x ẑ / r²
Here,
- μ₀ is the permeability of free space (4π x 10^-7 T·m/A).
- I_plane is the current density of the plane (100 mA/m).
- dl_plane is an infinitesimally small length element along the plane.
- ẑ is the unit vector pointing in the z direction.
- r is the distance from the plane to the point (4, 3, 2).
3. Since the plane is parallel to the yz-plane, the distance (r) between the plane and the point (4, 3, 2) will be the difference between the x-coordinates: r = 10 - 4 = 6.
4. The infinitesimally small length element (dl_plane) along the plane is in the az direction and can be written as dl_plane = dy_plane * az, where dy_plane is an infinitesimally small distance along the y-direction.
5. Substituting the values into the Biot-Savart law equation:
dH_plane = (4π x 10^-7 T·m/A) * (100 mA/m) * dy_plane * ẑ / 6²
6. Integrating the Biot-Savart law equation over the y-coordinate, from -∞ to +∞, gives the total contribution of the plane to the magnetic field at the point (4, 3, 2):
H_plane = ∫[ -∞ to +∞ ] (4π x 10^-7 T·m/A) * (100 mA/m) * dy_plane * ẑ / 6²
Simplifying the equation further, we get:
H_plane = (4π x 10^-7 T·m/A) * (100 mA/m) * ẑ / (6²) * ∫[ -∞ to +∞ ] dy_plane
7. The integral of dy_plane over the entire y-coordinate range is equal to the total length of the plane. Since the plane is infinite, the length is also infinite, resulting in a divergence. Thus, the integral is not finite. Therefore, the contribution to the magnetic field from the infinite plane is infinite.
8. Now, consider the line carrying current. The line x = 1, y = -2 carries a filamentary current of 20π mA along the az direction. To find the magnetic field at (4, 3, 2) due to this line, we can apply the Biot-Savart law again.
9. The contribution to the magnetic field at the point (4, 3, 2) from the line can be calculated using the Biot-Savart law as follows:
dH_line = (μ₀/4π) * I_line * dl_line x ẑ / r
Here,
- μ₀ is the permeability of free space (4π x 10^-7 T·m/A).
- I_line is the current of the line (20π mA).
- dl_line is an infinitesimally small length element along the line.
- r is the distance from the line to the point (4, 3, 2).
10. Since the line is not parallel to any coordinate axis, we need to calculate the distance (r) between the line and the point (4, 3, 2) by taking into account the coordinates.
11. The infinitesimally small length element (dl_line) along the line can be written as dl_line = dx_line * ax + dy_line * ay, where dx_line and dy_line are infinitesimally small distances along the x and y directions, respectively.
12. Substituting the values into the Biot-Savart law equation:
dH_line = (4π x 10^-7 T·m/A) * (20π mA) * (dx_line * ax + dy_line * ay) x ẑ / r
13. Integrating the Biot-Savart law equation over the entire length of the line will give us the total contribution of the line to the magnetic field at the point (4, 3, 2). However, since we do not have the range of integration for the x-coordinate of the line, we cannot calculate the exact contribution.
In summary, the magnetic field (H) at the point (4, 3, 2) due to the given current sources cannot be determined accurately because the contribution from the infinite plane is infinite, and the range of integration for the x-coordinate of the line is not specified.