A five-sided object, whose dimensions are shown in the figure, is placed in a uniform magnetic field. The magnetic field has a magnitude of 0.72 T and points along the positive y direction. Determine the magnetic flux through each of the five sides.
picture can be found on webassign (. net /) cj8/22-p-017 . gif
Front Triangle (yz plane at x = 1.2 m)
-Sides of
0.30 m, 0.40 m, 0.50 m
Triangle in back (yz plane at x = 0 m)
-Sides of
0.30 m, 0.40 m, 0.50 m
Rectangle on the left (in the xz plane)
-Sides of
0.3 m, 1.2 m
Rectangle on the top
-Sides of
0.5 m, 1.2 m
Rectangle on the bottom (in the xy plane)
-Sides of
0.4 m, 1.2 m
Well, well, well, we have ourselves a magnetic mystery here! Let's use our funny bones to solve it.
To determine the magnetic flux through each of the five sides, we need to use the formula:
Flux = magnetic field strength (B) * area (A) * cosine of the angle between the magnetic field and the surface normal.
So, let's calculate the magnetic flux through each side step by step, and I promise to sprinkle in some chuckles along the way!
1. Front Triangle (yz plane at x = 1.2 m):
The area of the triangle is (0.5 * 0.4) / 2 = 0.1 m². (You get the area by being a master at math formulas or by asking nicely.)
The angle between the magnetic field and the surface normal is 90 degrees because the field is pointing along the positive y direction and the triangle is perpendicular to it. So the cosine of 90 degrees is 0.
Therefore, the magnetic flux through the front triangle is 0 T*m². It's like trying to make a sandwich out of ghost bread. Not gonna work!
2. Back Triangle (yz plane at x = 0 m):
The area of this triangle is also (0.5 * 0.4) / 2 = 0.1 m². (Twice the chuckles!)
Again, the angle between the magnetic field and the surface normal is 90 degrees, so the cosine of 90 degrees is 0.
Therefore, the magnetic flux through the back triangle is also 0 T*m². Magnetic field: "I got 99 problems, but these triangles ain't one!"
3. Left Rectangle (xz plane):
The area of the rectangle is 0.3 * 1.2 = 0.36 m². (Let's not cut corners when calculating areas, okay?)
This time, the magnetic field is perpendicular to the rectangle, so the angle between them is 90 degrees. The cosine of 90 degrees is, you guessed it, 0.
Hence, the magnetic flux through the left rectangle is 0 T*m². It's like trying to find the humor in a math joke. Sometimes, it just doesn't add up!
4. Top Rectangle:
The area of the top rectangle is 0.5 * 1.2 = 0.6 m². (Don't look down on rectangles just because they're not triangles!)
The magnetic field is parallel to the top rectangle, so the angle between them is 0 degrees. The cosine of 0 degrees is 1. (Finally, a non-zero cosine!)
Therefore, the magnetic flux through the top rectangle is B * A * cos(0) = 0.72 T * 0.6 m² * 1 = 0.432 T*m². We've got a winner, folks!
5. Bottom Rectangle (xy plane):
The area of the bottom rectangle is 0.4 * 1.2 = 0.48 m². (No square business here, just plain old rectangles!)
Similar to the top rectangle, the magnetic field is parallel to the bottom rectangle, so the angle between them is 0 degrees. The cosine of 0 degrees is once again 1.
Thus, the magnetic flux through the bottom rectangle is B * A * cos(0) = 0.72 T * 0.48 m² * 1 = 0.3456 T*m². Not too shabby!
So, to sum it up:
- Front Triangle: 0 T*m²
- Back Triangle: 0 T*m²
- Left Rectangle: 0 T*m²
- Top Rectangle: 0.432 T*m²
- Bottom Rectangle: 0.3456 T*m²
And that's how the magnetic humor flows!
To determine the magnetic flux through each of the five sides, we can use the formula for magnetic flux:
Φ = B * A * cos(θ)
Where:
Φ = magnetic flux
B = magnetic field strength
A = area vector
θ = angle between the magnetic field and the area vector
Let's calculate the magnetic flux for each side step-by-step:
1) Front Triangle (yz plane at x = 1.2 m):
Area of the triangle (A) = (0.5 * base * height) = (0.5 * 0.4 m * 0.5 m) = 0.1 m^2
Angle between the magnetic field and the area vector (θ) = 0 degrees
Magnetic flux (Φ) = B * A * cos(θ) = 0.72 T * 0.1 m^2 * cos(0 degrees) = 0.072 T·m^2
2) Triangle in back (yz plane at x = 0 m):
Area of the triangle (A) = (0.5 * base * height) = (0.5 * 0.4 m * 0.5 m) = 0.1 m^2
Angle between the magnetic field and the area vector (θ) = 180 degrees
Magnetic flux (Φ) = B * A * cos(θ) = 0.72 T * 0.1 m^2 * cos(180 degrees) = -0.072 T·m^2
Note: The negative sign indicates that the magnetic flux is in the opposite direction.
3) Rectangle on the left (in the xz plane):
Area of the rectangle (A) = length * width = 0.3 m * 1.2 m = 0.36 m^2
Angle between the magnetic field and the area vector (θ) = 90 degrees
Magnetic flux (Φ) = B * A * cos(θ) = 0.72 T * 0.36 m^2 * cos(90 degrees) = 0 T·m^2
Note: The magnetic field is perpendicular to the area vector, so the magnetic flux is zero.
4) Rectangle on the top:
Area of the rectangle (A) = length * width = 0.5 m * 1.2 m = 0.6 m^2
Angle between the magnetic field and the area vector (θ) = 0 degrees
Magnetic flux (Φ) = B * A * cos(θ) = 0.72 T * 0.6 m^2 * cos(0 degrees) = 0.432 T·m^2
5) Rectangle on the bottom (in the xy plane):
Area of the rectangle (A) = length * width = 0.4 m * 1.2 m = 0.48 m^2
Angle between the magnetic field and the area vector (θ) = 90 degrees
Magnetic flux (Φ) = B * A * cos(θ) = 0.72 T * 0.48 m^2 * cos(90 degrees) = 0 T·m^2
So, the magnetic flux through each side is as follows:
Front Triangle: 0.072 T·m^2
Triangle in back: -0.072 T·m^2
Rectangle on the left: 0 T·m^2
Rectangle on the top: 0.432 T·m^2
Rectangle on the bottom: 0 T·m^2
To determine the magnetic flux through each of the five sides, we can use the formula:
Φ = B * A * cos(θ),
where Φ is the magnetic flux, B is the magnitude of the magnetic field, A is the area of the surface, and θ is the angle between the magnetic field and the surface normal.
Let's calculate the magnetic flux for each side:
1. Front Triangle (yz plane at x = 1.2 m):
- Side lengths: 0.30 m, 0.40 m, 0.50 m
- Area (A): (0.30 m) * (0.40 m) * 0.5 = 0.06 square meters
- Angle (θ): The magnetic field points along the positive y direction, so the angle between the magnetic field and the surface normal is 0 degrees.
- Magnetic flux (Φ): Φ = (0.72 T) * (0.06 square meters) * cos(0 degrees) = 0.0432 Weber (Wb)
2. Triangle in back (yz plane at x = 0 m):
- Side lengths: 0.30 m, 0.40 m, 0.50 m
- Area (A): (0.30 m) * (0.40 m) * 0.5 = 0.06 square meters
- Angle (θ): The magnetic field points along the positive y direction, so the angle between the magnetic field and the surface normal is 0 degrees.
- Magnetic flux (Φ): Φ = (0.72 T) * (0.06 square meters) * cos(0 degrees) = 0.0432 Weber (Wb)
3. Rectangle on the left (in the xz plane):
- Side lengths: 0.3 m, 1.2 m
- Area (A): (0.3 m) * (1.2 m) = 0.36 square meters
- Angle (θ): The magnetic field points along the positive y direction, so the angle between the magnetic field and the surface normal is 90 degrees.
- Magnetic flux (Φ): Φ = (0.72 T) * (0.36 square meters) * cos(90 degrees) = 0 Wb (Since cos(90 degrees) = 0)
4. Rectangle on the top:
- Side lengths: 0.5 m, 1.2 m
- Area (A): (0.5 m) * (1.2 m) = 0.6 square meters
- Angle (θ): The magnetic field points along the positive y direction, so the angle between the magnetic field and the surface normal is 0 degrees.
- Magnetic flux (Φ): Φ = (0.72 T) * (0.6 square meters) * cos(0 degrees) = 0.432 Weber (Wb)
5. Rectangle on the bottom (in the xy plane):
- Side lengths: 0.4 m, 1.2 m
- Area (A): (0.4 m) * (1.2 m) = 0.48 square meters
- Angle (θ): The magnetic field points along the positive y direction, so the angle between the magnetic field and the surface normal is 90 degrees.
- Magnetic flux (Φ): Φ = (0.72 T) * (0.48 square meters) * cos(90 degrees) = 0 Wb (Since cos(90 degrees) = 0)
Therefore, the magnetic flux through each of the five sides is as follows:
Front Triangle: 0.0432 Wb
Back Triangle: 0.0432 Wb
Left Rectangle: 0 Wb
Top Rectangle: 0.432 Wb
Bottom Rectangle: 0 Wb