Chapter 5: Q34P (page 248)

A dielectric lamina with charge density proportional to y covers the area between the parabola $y = 16 \cdot x^{2}$ and the x axis. Find the total charge.

Short Answer

Expert verified

The total charge of the dielectric lamina is, $546.13 K$ .

Step by step solution

Definition of double integral and mass formula

The double integral of $f (x, y)$ over the area Ain the (x,y) plane as the limit of this sum, and we write it as

Using the double integral the mass of the lamina:localid="1658895349906" $M = \int \int_{A} ρdA$ .

Diagram of the dielectric lamina

The figure of the dielectric lamina with charge density proportional to y covers the area between the parabola $y = 16 - x^{2}$ and x theaxis.

Calculation of the charge of the dielectric lamina

The total charge of the lamina is given by,

$Q = \int \int A ρdA = \int_{- 4}^{4} [\int_{0}^{16 - x^{2}} Kydy] dx = K \int_{- 4}^{4} ({\frac{1}{2} y^{2}}_{0}^{16 - x^{2}}) dx = \frac{K}{2} \int_{- 4}^{4} {(16 - x^{2})}^{2} dx = \frac{K}{2} \int_{- 4}^{4} (256 - 32 x^{2} + x^{4}) dx = \frac{K}{2} {[256 x - \frac{32}{3} x^{3} + \frac{1}{5} x^{5}]}_{- 4}^{4} = 546.13 K$

where K is the proportionality constant.

Therefore, the total charge is, $546.13 K$ .

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A dielectric lamina with charge density proportional to y covers the area between the parabola $y = 16 \cdot x^{2}$ and the x axis. Find the total charge.

Short Answer

Step by step solution

Definition of double integral and mass formula

Diagram of the dielectric lamina

Calculation of the charge of the dielectric lamina

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A dielectric lamina with charge density proportional to y covers the area between the parabola y=16·x2 and the x axis. Find the total charge.

Short Answer

Step by step solution

Definition of double integral and mass formula

Diagram of the dielectric lamina

Calculation of the charge of the dielectric lamina

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electromagnetism

Kinematics Physics

Energy Physics

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Study anywhere. Anytime. Across all devices.

A dielectric lamina with charge density proportional to y covers the area between the parabola $y = 16 \cdot x^{2}$ and the x axis. Find the total charge.