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Chapter 6: Q11P (page 295)

(a) Given $桅 = x^{2} - y^{2}$ , sketch on one graph the curves $桅 = 4, 桅 = 1, 桅 = 0, 桅 = - 1, 桅 = - 4$ . If $桅$ is the electrostatic potential, the curves $桅 =$ const. are equipotential, and the electric field is given by $E = - 鈭� 桅$ . If $桅$ is temperature, the curves $桅$ = const. are isothermals and $鈭� 桅$ is the temperature gradient; heat flows in the direction $- 鈭� 桅$ .
(b) Find and draw on your sketch the vectors $- 鈭� 桅$ at the points $(x, y) = (+ 1, + 1)$ , $(0, + 2), (+ 2, 0)$ ,. Then, remembering that $鈭� 桅$ is perpendicular to $桅$ = const., sketch, without computation, several curves along which heat would flow [see (a)].

Short Answer

Expert verified

Sketches are plotted as mentioned below.

(a)

(b)

Step by step solution

01

Given Information.

The value of the scalar field is $蠒 = x^{2} - y^{2}$ .

02

Definition of gradient.

Gradient is defined by the equation mentioned below

$鈭� 蠒 = \frac{鈭� 蠒}{鈭� x} \hat{i} + \frac{鈭� 蠒}{鈭� y} \hat{j} + \frac{鈭� 蠒}{鈭� z} \hat{k}$

03

Write down the equations for the given values of electrostatic potential.

(a) Write the first one.

$蠒 = 4 x^{2} - y^{2} = 4$

Write the second one.

$蠒 = 1 x^{2} - y^{2} = 1$

Write the third one.

$蠒 = 0 x^{2} - y^{2} = 0$

Write the fourth one.

$蠒 = - 1 x^{2} - y^{2} = - 1$

Write the fifth one.

$蠒 = - 4 x^{2} - y^{2} = - 4$

04

Sketch the plot of these equations.

The graph looks like as presented here.

05

Find the gradient of the function.

The gradient of the function becomes as shown below.

$鈭� 蠒 = \frac{未 (x^{2} - y^{2})}{未 x} i + \frac{未 (x^{2} - y^{2})}{未 y} j = 2 x i - 2 y j$

06

Calculate the values of gradients at different points.

(b) The first point is $(1, 1)$ .

Calculate the gradient.

$- 鈭� 蠒 = - 2 i + 2 j$

The second point is $(1, - 1)$ .

Calculate the gradient.

$- 鈭� 蠒 = - 2 i + 2 j$

The third point is $(- 1, 1)$ .

Calculate the gradient.

$- 鈭� 蠒 = - 2 i + 2 j$

The fourth point is $(- 1, - 1)$ .

Calculate the gradient.

$- 鈭� 蠒 = - 2 i + 2 j$

07

Sketch the plot.

Sketch the plot using the derived information.

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Most popular questions from this chapter

Given, integrate $V - n d 蟽$ over the whole surface of the cube of side 1 with four of its vertices at $(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0),$ Evaluate the same integral by means of the divergence theorem.

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