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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

What is the potential difference between $x_{i} = 10 c m$ and $x_{f} = 30 c m$ in the uniform electric field $E_{x} = 1000 V / m$ ?

What is the potential difference between $y_{i} = - 5 c m$ and $y_{f} = 5 c m$ in the uniform electric field $\vec{E} = (120, 000 \hat{i} - 50, 000 \hat{j}) V / m$ ?

Figure EX26.11 is a graph of $V versus x$ . Draw the corresponding graph of $E_{x} versus x$ .

Two $2.0 c m * 2.0 c m$ metal electrodes are spaced $1.0 m m$ apart and connected by wires to the terminals of a $9.0 V$ battery.

a. What are the charge on each electrode and the potential difference between them?

The wires are disconnected, and insulated handles are used to pull the plates apart to a new spacing of $2.0 m m$ .

b. What are the charge on each electrode and the potential difference between them?

The electric potential in a region of uniform electric field is $- 1000 V$ at $x = - 1.0 m$ and $+ 1000 V at x = + 1.0 m$ . What is $E_{x}$ ?

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