A 2.0 cm × 3.0 cm rectangle lies in the $xz$-plane with unit vector $\hat{n}$ pointing in the +y-direction. What is the electric flux through the rectangle if the electric field is $\overrightarrow{E}=(4000\hat{i}-2000\hat{k})$ N/C?

Verified step by step guidance

1. Step 1: Recall the formula for electric flux, which is Φ = E ⋅ A, where Φ is the electric flux, E is the electric field vector, and A is the area vector. The dot product accounts for the angle between the electric field and the area vector.
2. Step 2: Calculate the area of the rectangle. The dimensions are given as 2.0 cm × 3.0 cm. Convert these to meters (since SI units are required), so the area becomes A = (0.02 m)(0.03 m).
3. Step 3: Determine the area vector. The rectangle lies in the 𝓍𝒵-plane, and the unit vector nˆ points in the +y-direction. Therefore, the area vector A is perpendicular to the rectangle and has a magnitude equal to the area, with direction along the +y-axis: A = (0 î + A ĵ + 0 kˆ).
4. Step 4: Write the electric field vector E = (4000 î − 2000 kˆ) N/C. Note that the electric field has no component in the y-direction (ĵ).
5. Step 5: Compute the dot product E ⋅ A. Since the electric field has no component in the y-direction, the dot product E ⋅ A = (4000)(0) + (0)(A) + (−2000)(0). This simplifies to zero, meaning the electric flux through the rectangle is zero.

Key Concepts

Electric Flux

Electric flux is a measure of the electric field passing through a given area. It is calculated as the dot product of the electric field vector and the area vector, which is perpendicular to the surface. The formula for electric flux (Φ) is Φ = E · A, where E is the electric field and A is the area vector. The unit of electric flux is the volt-meter (V·m) or equivalently, newton-meters squared per coulomb (N·m²/C).

Area Vector

The area vector is a vector that represents the magnitude and direction of a surface area. For a flat surface, the area vector is perpendicular to the surface and its magnitude is equal to the area of the surface. In this case, the rectangle lies in the xz-plane, so the area vector points in the y-direction, consistent with the unit vector nˆ. The area of the rectangle is calculated as length times width, which is 2.0 cm × 3.0 cm.

Dot Product

The dot product is a mathematical operation that takes two vectors and returns a scalar. It is calculated as the product of the magnitudes of the two vectors and the cosine of the angle between them. In the context of electric flux, the dot product between the electric field vector and the area vector determines how much of the electric field passes through the area. If the vectors are perpendicular, the dot product is zero, indicating no flux through the surface.