Chapter 6: Q11P (page 323)

Given that $B = curl A$ , use the divergence theorem to show that over any closed surface is zero.

Short Answer

Expert verified

The solution of the integrals is $∮_{\partial V} B \times ndσ = 0$ .

Step by step solution

Given Information.

The given integral is $∮ B \times nd σ .$

Definition of Divergence’s Theorem.

Divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed. According to this theorem, the surface integral of a vector field over a closed surface, also known as the flux through the surface, equals the volume integral of the divergence over the region inside the surface.

Apply Divergence Theorem.

Use the divergence theorem.

$∮_{a} B \times n d σ = \int_{1} (\nabla \times B) d τ_{I}$

$B = \nabla \times A$

Then, the equation is $∮_{\partial v} B \times nd σ = \int_{V} \nabla \times (\nabla \times A) d τ$ true.

$∮_{\partial v} B n d σ = \int_{v} \nabla (\nabla \times A) d τ$

But, $\nabla \times (\nabla \times A) = 0$ everywhere .

$∮_{\partial v} B \times n d σ = 0$

Hence, the solution of the integrals is $∮_{\partial v} B \times nd σ = 0$ .

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Given that $B = curl A$ , use the divergence theorem to show that over any closed surface is zero.

Short Answer

Step by step solution

Given Information.

Definition of Divergence’s Theorem.

Apply Divergence Theorem.

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Given that B=curlA, use the divergence theorem to show that over any closed surface is zero.

Short Answer

Step by step solution

Given Information.

Definition of Divergence’s Theorem.

Apply Divergence Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Conservation of Energy and Momentum

Electric Charge Field and Potential

Radiation

Fields in Physics

Dynamics

Study anywhere. Anytime. Across all devices.

Given that $B = curl A$ , use the divergence theorem to show that over any closed surface is zero.