Chapter 10: Q19P (page 528)

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates $\nabla \times (r e_{θ}), \nabla (r c o s θ), \nabla . r$ .

Short Answer

Expert verified

The required values are mentioned below.

$\nabla \times (r {\hat{e}}_{θ}) = 2 r {\hat{e}}_{ϕ} \nabla (r \cos θ) = \cos θ {\hat{e}}_{r} - \sin θ {\hat{e}}_{ϕ} \nabla \cdot r = 3$

Step by step solution

Definition of spherical coordinates.

The coordinate system primarily utilized in three-dimensional systems is the spherical coordinates of the system. The spherical coordinate system is used to find the surface area in three-dimensional space.

Determining the value of ∇×(reθ),∇(rcosθ),∇.r

Findthe value of $\nabla \times (r {\hat{e}}_{θ})$ .

role="math" localid="1659261996986" $\nabla \times (r {\hat{e}}_{θ}) = - \frac{1}{r^{2} \sin θ \partial ϕ} (r^{2}) {\hat{e}}_{r} + \frac{1}{r} \frac{\partial}{\partial r} r^{2} {\hat{e}}_{ϕ} \nabla \times (r {\hat{e}}_{θ}) = 2 r {\hat{e}}_{ϕ}$

Findthe value of .

role="math" localid="1659262028625" $\nabla (r \cos θ) = \frac{\partial}{\partial r} (r \cos θ) {\hat{e}}_{r} + \frac{1}{r} \frac{\partial}{\partial θ} (r \cos θ) {\hat{e}}_{r} \nabla (r \cos θ) = \cos θ {\hat{e}}_{r} - \sin θ {\hat{e}}_{r}$

Findthe value of .

$\nabla \cdot r = \frac{1}{r^{2} s i n θ} \frac{\partial}{\partial r} (r^{2} s i n θ r) = 3$

The required values are mentioned below.

role="math" localid="1659262073399" $\nabla \times (r {\hat{e}}_{θ}) = 2 r {\hat{e}}_{ϕ} \nabla (r \cos θ) = \cos θ {\hat{e}}_{r} - \sin θ {\hat{e}}_{r} \nabla \cdot r = \frac{1}{r^{2} s i n θ} \frac{\partial}{\partial r} (r^{2} s i n θ r) = 3$

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Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates $\nabla \times (r e_{θ}), \nabla (r c o s θ), \nabla . r$ .

Short Answer

Step by step solution

Definition of spherical coordinates.

Determining the value of ∇×(reθ),∇(rcosθ),∇.r

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Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates ∇×(reθ),∇(rcosθ),∇.r.

Short Answer

Step by step solution

Definition of spherical coordinates.

Determining the value of ∇×(reθ),∇(rcosθ),∇.r

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

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Electric Charge Field and Potential

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Solid State Physics

Measurements

Study anywhere. Anytime. Across all devices.

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates $\nabla \times (r e_{θ}), \nabla (r c o s θ), \nabla . r$ .