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Chapter 26: Q. 73 (page 741)

Derive Equation 26.33 for the induced surface charge density on the dielectric in a capacitor.

Short Answer

Expert verified

By deriving equation 26.33 for the induced surface charge density on the dielectric in a capacitor we get $η_{i} = (1 - \frac{1}{κ}) η_{0}$ .

Step by step solution

01

Given Information

Equation 26.33:

$η_{induced} = η_{0} (1 - \frac{1}{κ})$

$η_{induced}$ ranges from nearly zero when $κ \approx 1$ to $\approx η_{0}$ when $κ ≫ 1$ .

02

Explanation

From expression 26.29 we have

$E_{i} = E_{0} - E .$

The induced field will be

$E_{i} = \frac{η_{i}}{ε_{0}} .$

Substitute expression 26.30 to 26.26,

$κ : = \frac{E_{0}}{E} \Rightarrow E_{0} = κ E_{0},$

Now find

localid="1648710646958" $E_{i} = E_{0} - E = E (κ - 1) = (\frac{κ - 1}{κ}) E_{0} .$

The induced charge density will be related to the induced field as

$η_{i} = E_{i} ε_{0}$

Substituting the previous result,

localid="1648710633563" $η_{i} = (\frac{κ - 1}{κ}) E_{0} ε_{0}$

Note that $E_{0} ε_{0} = η_{0}$ and rearrange the ratio, to find the result as in the book:

$η_{i} = (1 - \frac{1}{κ}) η_{0}$

03

Final Answer

Hence, the expression will be $η_{i} = (1 - \frac{1}{κ}) η_{0} .$

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Most popular questions from this chapter

An electric dipole at the origin consists of two charges q spaced apart along the y-axis.

a. Find an expression for the potential V(x, y) at an arbitrary point in the xy-plane. Your answer will be in terms of q, s, x, and y.

b. Use the binomial approximation to simplify your result from part a when s V x and s V y.

c. Assuming s V x and y, find expressions for Ex and Ey, the components of E u for a dipole.

d. What is the on-axis field E? Does your result agree with Equation 23.10?

e. What is the field E u on the bisecting axis? Does your result agree with Equation 23.11?

What are the magnitude and direction of the electric field at the dot in Figure EX26.8?

a. Use the methods of Chapter 25 to find the potential at distance $x$ on the axis of the charged rod shown in FIGURE P26.43.

b. Use the result of part a to find the electric field at distance $x$ on the axis of a rod

You need a capacitance of $50 m F$ , but you don’t happen to have a $50 m F$ capacitor. You do have a $30 m F$ capacitor. What additional capacitor do you need to produce a total capacitance of $50 m F$ ? Should you join the two capacitors in parallel or in series?

Find expressions for the equivalent capacitance of

(a) $N$ identical capacitors $C$ in parallel and

(b) $N$ identical capacitors $C$ in series.

See all solutions

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