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Chapter 26: Q. 42 (page 739)

Use the on-axis potential of a charged disk from Chapter 25 to find the on-axis electric field of a charged disk

Short Answer

Expert verified

The electric field of charged disk is $E (z) = - \frac{Q}{2 π \in_{0}} R^{2} \frac{z}{\sqrt{R^{2} + z^{2}}} - 1$

Step by step solution

01

Given information

We need to find the on-axis electric field of a charged disk

02

Simplify

The potential on the axis of a disk with charge $Q$ and radius $R$ at distance $z$ from it is given as

$V (z) = \frac{Q}{2 π \in_{0} R^{2}} \sqrt{R^{2} + z^{2} - z}$ .

The formula for electric field is

$E = - \frac{d V}{d r}$ ,

Where $r$ is the direction in which we're investigating the electric field. We have to take the negative of the derivative of the first expression with respect to $z$ . It is given as

$E = - \frac{d V}{d z} = - \frac{Q}{2 π \in_{0} R^{2}} \frac{d}{d z} \sqrt{R^{2} + z^{2} - z}$

If we perform the calculation correctly we get

$E (z) = - \frac{Q}{2 π \in_{0} R^{2}} \frac{z}{\sqrt{R^{2} + z^{2}}} - 1$

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Most popular questions from this chapter

The electric field in a region of space is $e X = - 1000 \times v / M$ , where $x$ is in meters.

a. Graph $E x$ versus $x$ over the region $- 1 m . . . \times . . . 1 m$ .

b. What is the potential difference between $x i = - 20 c m$ and $x f = 30 c m$ ?

Consider a uniformly charged sphere of radius R and total cAlC charge Q. The electric field $E_{out}$ outside the sphere $(r \geq R)$ is simply that of a point charge Q. In Chapter 24, we used Gauss's law to find that the electric field $E_{in}$ inside the sphere $(r \leq R)$ is radially outward with field strength

$E_{in} = \frac{1}{4 π ϵ_{0}} \frac{Q}{R^{3}} r$

a. The electric potential $V_{out}$ outside the sphere is that of a point charge Q. Find an expression for the electric potential $V_{in}$ at position r inside the sphere. As a reference, let $V_{in} = V_{out}$ at the surface of the sphere.

b. What is the ratio $V_{center} / V_{surface} ?$

c. Graph V versus r for 0 $\leq$ r $\leq$ 3 R.

Two flat, parallel electrodes $2.5 c m$ apart are kept at potentials of $20 V and 35 V$ . Estimate the electric field strength between them.

The flash unit in a camera uses a $3.0 V$ battery to charge a capacitor. The capacitor is then discharged through a flashlamp. The discharge takes $10 μ s$ , and the average power dissipated in the flashlamp is $10 W$ . What is the capacitance of the capacitor?

To what potential should you charge a $1.0 m F$ capacitor to store $1.0 J$ of energy ?

See all solutions

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