Question

If \(\mathcal{S}\) is a plane, and \(\vec{F}\) is always parallel to \(\mathcal{S},\) then the flux of \(\vec{F}\) across \(\mathcal{S}\) will be ________.

Short answer

Zero.

Step-by-step solution

Understand the Problem

We are given a plane \( \mathcal{S} \) and a vector field \( \vec{F} \) that is always parallel to this plane. We need to determine the flux of \( \vec{F} \) across \( \mathcal{S} \).

Define Flux of a Vector Field

The flux of a vector field \( \vec{F} \) through a surface \( \mathcal{S} \) is given by the surface integral: \( \iint_{\mathcal{S}} \vec{F} \cdot \vec{n} \, dS \), where \( \vec{n} \) is the unit normal vector to the surface. This measures the quantity of \( \vec{F} \) passing through \( \mathcal{S} \).

Analyze the Direction of Vector Field

Since \( \vec{F} \) is always parallel to the plane \( \mathcal{S} \), it means that \( \vec{F} \) and \( \vec{n} \), the normal vector to the plane, are orthogonal.

Calculate the Dot Product

The dot product \( \vec{F} \cdot \vec{n} \) will be zero because \( \vec{F} \) is parallel to \( \mathcal{S} \) and any vector parallel to the plane is perpendicular to the normal vector. Therefore, \( \iint_{\mathcal{S}} 0 \, dS = 0 \).

Conclude the Flux Result

Since the dot product of \( \vec{F} \) and \( \vec{n} \) is zero over the entire surface, the flux of \( \vec{F} \) across \( \mathcal{S} \) is always zero.

Key concepts

Vector Field

In mathematics and physics, a vector field is a function that assigns a vector to each point in space. Imagine a windy day, where at each point in the park, there is a specific wind direction and speed. Here, the wind can be thought of as a vector field. A vector field can be represented in a three-dimensional space with a collection of vectors. These vectors may vary in magnitude and direction depending on the location in the field.

Vector fields play a crucial role in understanding various physical phenomena. They are frequently used in electromagnetics, fluid dynamics, and other sciences. Some common examples include magnetic fields, electric fields, and gravitational fields.

• Magnitude: The length of each vector, representing the intensity or strength.
• Direction: Shows the path along which vector quantities act at a particular point.
• Continuous: A smoothly changing field allows for gradient calculations and applications in calculus.

Surface Integral

A surface integral is a tool used to compute the integration of a vector field across a surface. It's like adding up all the tiny contributions of the field along a given shape or surface. Imagine draping a sheet over a curved object and observing how the vector field behaves across every point on the sheet.

To compute a surface integral of a vector field \( \vec{F} \) over a surface \( \mathcal{S} \), one must consider both the magnitude and the direction of the field relative to the surface. The process involves breaking the surface into infinitesimally small patches, calculating the contribution of \( \vec{F} \) on each patch, and summing them up.

• Applications: Used in physics to calculate quantities like flux through surfaces, sound across barriers, and heat exchange.
• Orientation: Vital in determining whether the field contributes positively or negatively across the surface.

Calculating a surface integral requires knowing the orientation of the surface, and it often involves unit normal vectors to discern the angle of intersection between \( \vec{F} \) and \( \mathcal{S} \).

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the context of vector fields and surface integrals, the dot product is used to project the vector field onto the surface by comparing directional alignment.

Mathematically, the dot product of two vectors \( \vec{A} = (a_1, a_2, a_3) \) and \( \vec{B} = (b_1, b_2, b_3) \) is given by:\[\vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3.\]This provides a measure of how aligned the two vectors are. Key properties of the dot product include:

• Commutative: \( \vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A} \).
• Distributive: \( \vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C} \).
• Orthogonality: Dot product is zero when vectors are perpendicular.

In the context of flux, if a field \( \vec{F} \) is parallel to the plane such that it is orthogonal to the normal vector, the dot product ends up zero, canceling any contributions in the flux calculation.

Unit Normal Vector

A unit normal vector is a vector that is perpendicular to a given surface at a specific point, with a length of one unit. This vector plays a crucial role in surface integrals and calculating the flux of a vector field across a surface.

In the context of the exercise, the unit normal vector \( \vec{n} \) to a surface is key because it helps define the orientation of the surface. It's like a tiny arrow that points directly away from the surface, aiding in the measurement of how much of the vector field flows through the surface, rather than along it.

• Normalization: A process to convert a vector into a unit vector by dividing it by its magnitude.
• Direction: Determines the orientation of a surface for positive or negative flux calculations.
• Applications: Used to simplify mathematical expressions in physics and engineering problems.

The unit normal vector is essential for creating the dot product in surface integrals. When the vector field is parallel to the surface, the normal vector ensures that the dot product with the field is zero, indicating no flux through the surface.