Chapter 6: Q17P (page 295)

Find, $\nabla r$ , where $r = \sqrt{x^{2} + y^{2}}$ , using (6.7) and also using (6.3). Show that your results are the same by using (4.11) and (4.12).

Short Answer

Expert verified

The solution to this problem is $\nabla r = e_{r}$ .

Step by step solution

Given Information

The given information is $r = \sqrt{x^{2} + y^{2}}$

Definition of Scalar field.

In mathematics and physics, scalar field or scalar-valued function referred to a scalar value to everypointin aspace– possiblyphysical space. The scalar may either be a (dimensionless)mathematical numberor aphysical quantity.

Find the solution.

Use the following equation:

$\nabla f = e_{r} \frac{\partial f}{\partial r} + e_{θ} \frac{1}{r} \frac{\partial f}{\partial θ} + e_{z} \frac{\partial f}{\partial z} \nabla r = \frac{a r}{a r} e_{r} = e_{r}$

Use the following equation:

$\nabla ϕ = g r a d ϕ = i \frac{\partial ϕ}{\partial x} + j \frac{\partial ϕ}{\partial y} + k \frac{\partial ϕ}{\partial z} .$

Find the value of r.

$r = \sqrt{x^{2} + y^{2}} . \nabla r = \frac{1}{\sqrt{x^{2} + y^{2}}} (x i + y j) = \cos θ i + \sin θ = e_{r}$

Hence, by using both the equation (1) and (2), the answer is same $\nabla r = e_{r}$ .

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Find, $\nabla r$ , where $r = \sqrt{x^{2} + y^{2}}$ , using (6.7) and also using (6.3). Show that your results are the same by using (4.11) and (4.12).

Short Answer

Step by step solution

Given Information

Definition of Scalar field.

Find the solution.

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Most popular questions from this chapter

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Find, ∇r, wherer=x2+y2 , using (6.7) and also using (6.3). Show that your results are the same by using (4.11) and (4.12).

Short Answer

Step by step solution

Given Information

Definition of Scalar field.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Wave Optics

Physical Quantities And Units

Circular Motion and Gravitation

Fluids

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

Find, $\nabla r$ , where $r = \sqrt{x^{2} + y^{2}}$ , using (6.7) and also using (6.3). Show that your results are the same by using (4.11) and (4.12).