Chapter 6: Q20P (page 295)

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent $\nabla r^{2}$

Short Answer

Expert verified

$\nabla r^{2} = 2 r e_{r}$

Step by step solution

Given Information

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

Definition of Scalar field.

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

Finding the gradient

Using the equation,

$\begin{array}{rcl} \nabla ϕ & = & g r a d ϕ \\ = & i \frac{\partial ϕ}{\partial x} + j \frac{\partial ϕ}{\partial y} + k \frac{\partial ϕ}{\partial z} \end{array}$

Find the value of gradient.

$\begin{array}{rcl} r^{2} & = & x^{2} + y^{2} \\ \nabla r^{2} & = & 2 x \hat{i} + 2 y \hat{j} \end{array}$

$\begin{array}{rcl} \nabla r^{2} & = & 2 x \times \frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}} \hat{i} + 2 y \times \frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}} \hat{j} \\ = & 2 \sqrt{x^{2} + y^{2}} [\frac{x}{\sqrt{x^{2} + y^{2}}} \hat{i} + \frac{y}{\sqrt{x^{2} + y^{2}}} \hat{j}] \\ = & 2 r [\cos (θ) \hat{i} + s i n (θ) \hat{j}] \\ = & 2 r e_{r} \end{array}$

Hence, the solution to this problem is $\nabla r^{2} = 2 r e_{r}$ .

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As in Problem 17, find the following gradients in two ways and show that your answers are equivalent $\nabla r^{2}$

Short Answer

Step by step solution

Given Information

Definition of Scalar field.

Finding the gradient

One App. One Place for Learning.

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As in Problem 17, find the following gradients in two ways and show that your answers are equivalent ∇r2

Short Answer

Step by step solution

Given Information

Definition of Scalar field.

Finding the gradient

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Force

Conservation of Energy and Momentum

Rotational Dynamics

Space Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

As in Problem 17, find the following gradients in two ways and show that your answers are equivalent $\nabla r^{2}$