Chapter 6: Q2P (page 313)

$\oint 2 x d y - 3 y d x$ around the square with vertices $(0, 2), (2, 0), (- 2, 0), (0, - 2) .$

Short Answer

Expert verified

The integral will give the solution to be 40.

Step by step solution

Given information

The vertices of the square are $(0, 2), (2, 0), (- 2, 0, (0, - 2)$

Definition of conservative force and scalar potential

A force is said to be conservative if $\nabla \times F = 0 .$ .

The scalar potential is independent of the path. The scalar potential is the sum of potential in all the 3 dimensions calculated separately.

The formula for the scalar potential is $W = \int F . d r .$ .

Use substitution.

The green theorem in a plane is mentioned.

$\iint \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) = \oint P d x + Q d y$

Takethe values as mentioned below.

$P = - 3 y Q = 2 x$

Substitute the value.

$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2 + 3 \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 5$

Use Greens Theorem.

Use the Green theorem.

$- 3 x d x + 2 y d y 5 \iint d x d y 5 \int_{0}^{\sqrt{8}} \int_{0}^{\sqrt{8}} d x d y 5 \int_{0}^{\sqrt{8}} \sqrt{8} d x 5 \sqrt{8} \sqrt{8} 40$

Hence the integral $\oint 2 x d y - 3 y d x$ gives 40

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$\oint 2 x d y - 3 y d x$ around the square with vertices $(0, 2), (2, 0), (- 2, 0), (0, - 2) .$

Short Answer

Step by step solution

Given information

Definition of conservative force and scalar potential

Use substitution.

Use Greens Theorem.

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∮2xdy-3ydxaround the square with vertices(0,2),(2,0),(-2,0),(0,-2).

Short Answer

Step by step solution

Given information

Definition of conservative force and scalar potential

Use substitution.

Use Greens Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

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Study anywhere. Anytime. Across all devices.

$\oint 2 x d y - 3 y d x$ around the square with vertices $(0, 2), (2, 0), (- 2, 0), (0, - 2) .$