Q3.14P For the infinite slot (Ex. 3.3),... [FREE SOLUTION]

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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 3: Q3.14P (page 140)

For the infinite slot (Ex. 3.3), determine the charge density $蟽 (y)$ on the strip at $x = 0$ , assuming it is a conductor at constant potential $v_{0}$ .

Short Answer

Expert verified

The expression for the charge density on the strip at $x = 0$ is $蟽 (y) = \frac{4 蔚_{0} V_{0}}{a} \underset{n = 1, 3 .5 鈥�}{鈭�} 鈥� \sin 鈦� (\frac{苍蟺测}{a})$ .

Step by step solution

Define functions

Write the expression for the potential $V (x, y)$ in the infinite slot.

$V (x, y) = \frac{4 V_{0}}{蟺} \underset{n = 1, 3, 5, 鈥� .}{鈭�} 鈥� \frac{1}{n} e^{鈭� 苍蟺虫} \sin 鈦� \frac{苍蟺测}{a}$ 鈥︹€� (1)

Here, $v_{0}$ is the constant potential along the conductor, $x$ is the x-coordinate, $y$ is the y-coordinate and is the positive integer.

Determine charge density

Derive the charge density in the terms of electric potential.

$蟽 = 鈭� 蔚_{0} \frac{鈭� V}{鈭� n}$

$蟽 (y) = 鈭� 蔚_{0} {[\frac{鈭� V}{鈭� x}]}_{x 鈭� 0}$ 鈥︹€� (2)

Substitute $\frac{4 V_{0}}{蟺} \underset{n}{鈭�} 鈥� \frac{1}{n} e^{鈭� n_{i, 5}} \sin 鈦� \frac{苍蟺测}{a}$ for $V (x, y)$ in equation (2).

$蟽 (y) = 鈭� {蔚_{0} \frac{鈭�}{鈭� x} \{\frac{4 V_{0}}{蟺} 鈭� \frac{1}{n} e^{\frac{苍蟺虫}{a}} \sin 鈦� (\frac{苍蟺测}{a})\}|}_{x 鈭� 0}$

$= 鈭� {蔚_{0} \frac{4 V_{0}}{蟺} \frac{鈭�}{鈭� x} \{鈭� \frac{1}{n} e^{苍蟺虫} \sin 鈦� (\frac{苍蟺测}{a})\}|}_{x = 0}$

$= {蔚_{0} \frac{4 V_{0}}{蟺} \frac{1}{n} (\frac{苍蟺}{a}) 鈭� e^{苍蟺虫} \sin 鈦� (\frac{苍蟺测}{a})|}_{x = 0}$

$蟽 (y) = \frac{4 蔚_{0} V_{0}}{a} \underset{n 鈭� 1, 3, 5 鈥�}{鈭�} 鈥� \sin 鈦� (\frac{苍蟺测}{a})$

Hence, the expression for the charge density on the strip at $x = 0$ is $蟽 (y) = \frac{4 蔚_{0} V_{0}}{a} \underset{n 鈭� 1, 3, 5 鈥�}{鈭�} 鈥� \sin 鈦� (\frac{苍蟺测}{a})$ .

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91影视

For the infinite slot (Ex. 3.3), determine the charge density $蟽 (y)$ on the strip at $x = 0$ , assuming it is a conductor at constant potential $v_{0}$ .

Short Answer

Step by step solution

Define functions

Determine charge density

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91影视

For the infinite slot (Ex. 3.3), determine the charge density 蟽(y) on the strip at x=0, assuming it is a conductor at constant potential v0.

Short Answer

Step by step solution

Define functions

Determine charge density

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fluids

Solid State Physics

Famous Physicists

Astrophysics

Electrostatics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

For the infinite slot (Ex. 3.3), determine the charge density $蟽 (y)$ on the strip at $x = 0$ , assuming it is a conductor at constant potential $v_{0}$ .