Question

Given that $\mathbf{B}=$ curl $\mathbf{A},$ use the divergence theorem to show that $\oint \mathbf{B}\cdot \mathbf{n}d\sigma$ over any closed surface is zero.

Short answer

By applying the Divergence Theorem and the identity $abla\cdot (abla\times \mathbf{A})=0$, the surface integral $\oint \mathbf{B}\cdot \mathbf{n}d\sigma$ is zero.

Step-by-step solution

- State the Divergence Theorem

The Divergence Theorem states that for any vector field $\mathbf{F}$, the surface integral of the flux over a closed surface $S$ is equal to the volume integral of the divergence of $\mathbf{F}$ over the region $V$ bounded by $S$: $${\oint }_S\mathbf{F}\cdot \mathbf{n}d\sigma ={\iiint }_{V}abla\cdot \mathbf{F}dV$$

- Apply the Divergence Theorem to $\mathbf{B}$

Given $\mathbf{B}=abla\times \mathbf{A}$, we can write the surface integral of $\mathbf{B}$ over $S$ as: $${\oint }_S\mathbf{B}\cdot \mathbf{n}d\sigma ={\iiint }_{V}abla\cdot \mathbf{B}dV$$

- Use the Identity for Divergence of Curl

Recall the vector identity that states the divergence of the curl of any vector field is always zero: $$abla\cdot (abla\times \mathbf{A})=0$$

- Conclude the Solution Using the Identity

By applying this identity to our expression from Step 2, we get: $${\iiint }_{V}abla\cdot \mathbf{B}dV={\iiint }_{V}0dV=0$$ Since the volume integral of zero is zero, it follows that the original surface integral is also zero: $${\oint }_S\mathbf{B}\cdot \mathbf{n}d\sigma =0$$

Key concepts

Vector Field

In mathematics and physics, a vector field is a function that assigns a vector to each point in a space. Imagine a vector field as a set of arrows pointing in various directions in a three-dimensional space. Each arrow has a magnitude and a direction, representing the influence at that specific point.

Common examples of vector fields include:

• Gravitational fields - where vectors point towards the source of gravity.
• Electric and magnetic fields - where vectors can vary in strength and direction.
• Fluid flow fields - showing the velocity of particles in a fluid.

Understanding vector fields is crucial for comprehending further concepts like curl and divergence.

Divergence

Divergence is a measure of the magnitude of a field's source or sink at a given point. Essentially, it indicates how much a vector field is expanding or converging at a specific location. The mathematical notation for divergence is $abla\cdot \mathbf{F}$, where $\mathbf{F}$ represents the vector field.

Here's how you can understand divergence better:

• If $abla\cdot \mathbf{F}>0$, the field is diverging from that point (like air flowing out from a balloon).
• If $abla\cdot \mathbf{F}<0$, the field is converging to that point (like air being sucked into a vacuum).
• If $abla\cdot \mathbf{F}=0$, there is no net flow in or out, and the field could be circulating or steady in that region.

The divergence theorem connects the surface integral of a vector field to the volume integral of its divergence within that surface.

Curl

The curl of a vector field represents the rotation at a point within the field. It measures how much and in what direction the field 'swirls' around a point. Mathematically, the curl is denoted by $abla\times \mathbf{A}$, where $\mathbf{A}$ is the vector field.

Key points to understand about curl:

• If the curl of a vector field is zero, it means there is no rotation at that point – the fluid may be moving, but not swirling.
• A non-zero curl indicates a rotating motion around that point (imagine a whirlpool).

For instance, in electromagnetism, the curl of the electric field is related to the changing magnetic field. The identity $abla\cdot (abla\times \mathbf{A})=0$ tells us that the divergence of the curl of any vector field is always zero.

Surface Integral

A surface integral is a generalization of multiple integrals to integrate over surfaces. It allows for calculating the flux through a surface. If $\mathbf{F}$ is a vector field and $S$ is a surface, then the surface integral is represented as ${\oint }_S\mathbf{F}\cdot \mathbf{n}d\sigma$, where $\mathbf{n}$ is the unit normal vector to the surface.

When performing surface integrals, keep in mind:

• They are used to calculate flux across surfaces.
• The setup depends on the surface's shape and the vector field's complexity.
• They're essential in physics for concepts like electromagnetic fields and fluid flow.

The divergence theorem simplifies surface integrals by converting them to volume integrals, facilitating easier computations.