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Chapter 6: Problem 11

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

Expert verified
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

01

- 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 \(S\): \[ \oint_S \mathbf{F} \cdot \mathbf{n} \, d \sigma = \iiint_V abla \cdot \mathbf{F} \, dV \]
02

- 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: \[ \oint_S \mathbf{B} \cdot \mathbf{n} \, d \sigma = \iiint_V abla \cdot \mathbf{B} \, dV \]
03

- 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: \[ abla \cdot (abla \times \mathbf{A}) = 0 \]
04

- Conclude the Solution Using the Identity

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(abla \cdot \mathbf{F} \gt 0\), the field is diverging from that point (like air flowing out from a balloon).
  • If \(abla \cdot \mathbf{F} \lt 0\), 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 鈥� 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.

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Most popular questions from this chapter

Prove that the triple scalar product of \((\mathbf{A} \times \mathbf{B}),(\mathbf{B} \times \mathbf{C}),\) and \((\mathbf{C} \times \mathbf{A}),\) is equal to the square of the triple scalar product of \(\mathbf{A}, \mathbf{B},\) and \(\mathbf{C} .\) Hint: First let \((\mathbf{B} \times \mathbf{C})=\mathbf{D},\) and evaluate \((\mathbf{A} \times \mathbf{B}) \times \mathbf{D} .\) [See Am. J. Phys. 66, 739 (1998).]

Evaluate each integral in the simplest way possible. \(\iint(\nabla \times \mathbf{V}) \cdot \mathbf{n} d \sigma\) over the surface consisting of the four slanting faces of a pyramid whose base is the square in the \((x, y)\) plane with corners at (0,0),(0,2),(2,0),(2,2) and whose top vertex is at \((1,1,2),\) where $$\mathbf{V}=\left(x^{2} z-2\right) \mathbf{i}+(x+y-z) \mathbf{j}-x y z \mathbf{k}$$

For the force ficld \(\mathbf{F}=-y \mathbf{i}+x \mathbf{j}+z \mathbf{k},\) calculate the work done in moving a particle from (1,0,0) to \((-1,0, \pi)\). (a) along the helix \(x=\cos t, y=\sin t, z=t\) (b) along the straight line joining the points. Do you expect your answers to be the same? Why or why not?

The purpose in doing the following simple problems is to become familiar with the formulas we have discussed. So a good study method is to do them by hand and then check your results by computer. Compute the divergence and the curl of each of the following vector fields. $$\mathbf{V}=x^{2} y \mathbf{i}+y^{2} x \mathbf{j}+x y z \mathbf{k}$$

Given \(\mathbf{A}=\mathbf{i}+\mathbf{j}-2 \mathbf{k}, \mathbf{B}=2 \mathbf{i}-\mathbf{j}+3 \mathbf{k}, \mathbf{C}=\mathbf{j}-5 \mathbf{k}\): Let \(O\) be the tail of \(\mathbf{B}\) and let \(\mathbf{A}\) be a force acting at the head of \(\mathbf{B}\). Find the torque of \(\mathbf{A}\) about \(O ;\) about a line through \(O\) perpendicular to the plane of \(\mathbf{A}\) and \(\mathbf{B}\); about a line through \(O\) parallel to \(\mathbf{C}\).
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