An electric field of intensity 4.27 kN/C is applied along the x-axis. Calculate the electric flux through a rectangular plane 0.350 m wide and 0.700 m long if the following conditions are true.
(a) The plane is parallel to the yz-plane.
(b) The plane is parallel to the xy-plane.
(c) The plane contains the y-axis, and its normal makes an angle of 41.7° with the x-axis.
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To calculate the electric flux, we can use the formula:
Φ = E * A * cos(θ)
Where:
Φ is the electric flux.
E is the electric field intensity.
A is the area of the plane.
θ is the angle between the electric field and the normal to the plane.
(a) The plane is parallel to the yz-plane, so the normal to the plane is in the x-direction. The angle between the electric field and the normal is 0°, so θ = 0°.
Φ(a) = E * A * cos(θ)
= (4.27 kN/C) * (0.350 m * 0.700 m) * cos(0°)
The cos(0°) is equal to 1, so we can simplify the equation:
Φ(a) = (4.27 kN/C) * (0.350 m * 0.700 m)
= 1.269 kNm²/C
Therefore, the electric flux through the rectangular plane is 1.269 kNm²/C.
(b) The plane is parallel to the xy-plane, so the normal to the plane is in the z-direction. The angle between the electric field and the normal is 90°, so θ = 90°.
Φ(b) = E * A * cos(θ)
= (4.27 kN/C) * (0.350 m * 0.700 m) * cos(90°)
The cos(90°) is equal to 0, so the equation becomes:
Φ(b) = (4.27 kN/C) * (0.350 m * 0.700 m) * 0
= 0
Therefore, the electric flux through the rectangular plane is 0.
(c) The plane contains the y-axis, and its normal makes an angle of 41.7° with the x-axis. The angle between the electric field and the normal is 41.7°, so θ = 41.7°.
Φ(c) = E * A * cos(θ)
= (4.27 kN/C) * (0.350 m * 0.700 m) * cos(41.7°)
Calculating the value:
Φ(c) = (4.27 kN/C) * (0.350 m * 0.700 m) * cos(41.7°)
≈ 1.632 kNm²/C
Therefore, the electric flux through the rectangular plane is approximately 1.632 kNm²/C.
To calculate the electric flux through a surface, we use the formula:
Electric Flux (Φ) = Electric Field (E) * Area (A) * Cos(θ)
where Φ is the electric flux, E is the electric field intensity, A is the area of the surface, and θ is the angle between the electric field and the normal to the surface.
(a) The plane is parallel to the yz-plane:
Since the plane is parallel to the yz-plane, the normal to the surface is perpendicular to the x-axis. Therefore, the angle θ between the electric field and the normal is 90 degrees.
Electric Flux (Φ) = Electric Field (E) * Area (A) * Cos(θ)
Φ = 4.27 kN/C * (0.350 m * 0.700 m) * Cos(90°)
Φ = 4.27 kN/C * 0.245 m² * 0
Φ = 0
Therefore, the electric flux through the rectangular plane is zero when it is parallel to the yz-plane.
(b) The plane is parallel to the xy-plane:
Since the plane is parallel to the xy-plane, the normal to the surface is perpendicular to the z-axis. Therefore, the angle θ between the electric field and the normal is 0 degrees.
Electric Flux (Φ) = Electric Field (E) * Area (A) * Cos(θ)
Φ = 4.27 kN/C * (0.350 m * 0.700 m) * Cos(0°)
Φ = 4.27 kN/C * 0.245 m² * 1
Φ = 1.04555 kN·m²/C
Therefore, the electric flux through the rectangular plane is 1.04555 kN·m²/C when it is parallel to the xy-plane.
(c) The plane contains the y-axis, and its normal makes an angle of 41.7° with the x-axis:
In this case, the angle θ between the electric field and the normal is 41.7 degrees.
Electric Flux (Φ) = Electric Field (E) * Area (A) * Cos(θ)
Φ = 4.27 kN/C * (0.350 m * 0.700 m) * Cos(41.7°)
Φ = 4.27 kN/C * 0.245 m² * 0.741
Φ = 0.74563 kN·m²/C
Therefore, the electric flux through the rectangular plane is 0.74563 kN·m²/C when it contains the y-axis and its normal makes an angle of 41.7° with the x-axis.