Q12P For Problem 11,聽(a) Find the ma... [FREE SOLUTION]

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Chapter 6: Q12P (page 295)

For Problem 11,
(a) Find the magnitude and direction of the electric field at (2,1).
(b) Find the direction in which the temperature is decreasing most rapidly at(-3,2)
(c) Find the rate of change of temperature with distance at (1,2)in the direction

Short Answer

Expert verified

(a) Magnitude: $2 \sqrt{5}$

Direction: $4 \hat{i} - 2 \hat{j}$

(b) Direction: $6 \hat{i} + 4 \hat{j}$

Step by step solution

Given Information.

The gradient is $鈭� 桅 = 2 x \hat{i} - 2 y \hat{j}$

Definition of gradient.

Gradient is defined by the equation mentioned below.

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

Find electric field by taking the negative of electric field.

Find the electric field using the formula.

$\begin{array}{rcl} E & = & - 鈭� 蠒 \\ - 鈭� 蠒 & = & - 2 x \hat{i} + 2 y \hat{j} \\ E & = & - 4 \hat{i} + 2 \hat{j} \\ | E | & = & 2 \sqrt{5} \end{array}$

Take the opposite direction of the gradient.

At (3,2), the direction of maximum decrease of temperature is in the opposite direction of its gradient.

$\begin{array}{rcl} 鈭� 桅 & = & 2 (- 3) \hat{i} - 2 (2) \hat{j} \\ \overset{鈫�}{v} & = & - 鈭� 桅 \\ \overset{鈫�}{v} & = & 6 \hat{i} + 4 \hat{j} \end{array}$

Take the vector dot product to find the rate of change of temperature.

In the direction $3 \hat{i} - \hat{j}$ , the rate of change of temperature with distance (1,2) is given by the dot product.

$\begin{array}{rcl} \frac{d 蠒}{d u} & = & (鈭� 蠒) 路 u \\ \hat{u} & = & \frac{1}{\sqrt{10}} (3, - 1) \\ 鈭� 蠒 & = & 2 \hat{i} - 4 \hat{j} \\ \frac{d 蠒}{d u} & = & \frac{1}{\sqrt{10}} (3, - 1) 路 (2, - 4) \\ \frac{d 蠒}{d u} & = & \sqrt{10} \end{array}$

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91影视

For Problem 11,
(a) Find the magnitude and direction of the electric field at (2,1).
(b) Find the direction in which the temperature is decreasing most rapidly at(-3,2)
(c) Find the rate of change of temperature with distance at (1,2)in the direction

Short Answer

Step by step solution

Given Information.

Definition of gradient.

Find electric field by taking the negative of electric field.

Take the opposite direction of the gradient.

Take the vector dot product to find the rate of change of temperature.

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Most popular questions from this chapter

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91影视

For Problem 11,(a) Find the magnitude and direction of the electric field at (2,1).(b) Find the direction in which the temperature is decreasing most rapidly at(-3,2)(c) Find the rate of change of temperature with distance at (1,2)in the direction

Short Answer

Step by step solution

Given Information.

Definition of gradient.

Find electric field by taking the negative of electric field.

Take the opposite direction of the gradient.

Take the vector dot product to find the rate of change of temperature.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Dynamics

Radiation

Energy Physics

Fundamentals of Physics

Torque and Rotational Motion

Study anywhere. Anytime. Across all devices.

For Problem 11,
(a) Find the magnitude and direction of the electric field at (2,1).
(b) Find the direction in which the temperature is decreasing most rapidly at(-3,2)
(c) Find the rate of change of temperature with distance at (1,2)in the direction