Chapter 3: Q14P (page 140)

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

Short Answer

Expert verified

Answer

The equation for the charge density on the strip at $x = 0$ is $σ (y) = \frac{4 ε_{0} V_{0}}{a} \sum_{n = 1, 3, 5 . . .} \sin (\frac{n πy}{a})$ .

Step by step solution

Define functions

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

$V (x, y) = \frac{4 V_{0}}{π} \sum_{n = 1, 3, 5 . . .} \frac{1}{n} e^{\frac{- n πx}{a}} s i n (\frac{n π y}{a})$ …… (1)

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

Determine the charge density

Derive the charge density in terms of electric potential.

$σ = - e_{0} \frac{\partial V}{\partial n} σ (y) = - e_{0} {[\frac{\partial V}{\partial x}]}_{x = 0}$ …… (2)

Substitute $\frac{4 V_{0}}{π} \sum_{n = 1, 3, 5 . . .} \frac{1}{n} e^{\frac{- n πx}{a}} \sin \frac{n πy}{a}$ for $V (x, y)$ in equation (2).

${σ (y) = ε_{0} \frac{\partial}{\partial x} \{\frac{4 V_{0}}{π} \underset{}{\sum \frac{1}{n} e^{\frac{n πx}{a}}} \sin (\frac{n πy}{a})\}}_{x = 0} {= ε_{0} \frac{4 V_{0}}{π} \frac{\partial}{\partial x} \{\underset{}{\sum \frac{1}{n} e^{\frac{n πx}{a}}} \sin (\frac{n πy}{a})\}}_{x = 0} {= ε_{0} \frac{4 V_{0}}{π} \sum \frac{1}{n} \{(- \frac{n x}{a}) \underset{}{e^{\frac{n πx}{a}}} \sin (\frac{n πy}{a})\}}_{x = 0} {= ε_{0} \frac{4 V_{0}}{π} \frac{1}{n} (\frac{n x}{a}) \underset{}{\sum \frac{1}{n} e^{\frac{n πx}{a}}} \sin (\frac{n πy}{a})}_{x = 0}$

$σ (y) = \frac{4 ε_{0} V_{0}}{a} \sum_{n = 1, 3, 5 . . .} \sin (\frac{n πy}{a})$

Hence, the equation for the charge density on the strip at $x = 0$ is $σ (y) = \frac{4 ε_{0} V_{0}}{a} \sum_{n = 1, 3, 5 . . .} \sin (\frac{n πy}{a})$ .

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For the infinite slot (Ex. 3.3), determine the charge density $σ (y)$ 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 the charge density

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For the infinite slot (Ex. 3.3), determine the charge density σ(y)onthe strip at x=0, assuming it is a conductor at constant potential V0.

Short Answer

Step by step solution

Define functions

Determine the charge density

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Particle Model of Matter

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Electrostatics

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Study anywhere. Anytime. Across all devices.

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