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Chapter 23: Q. 46 (page 655)

A ring of radius $R$ has total charge $Q$ .
$(a)$ At what distance along the $z$ -axis is the electric field strength a maximum?
$(b)$ What is the electric field strength at this point?

Short Answer

Expert verified

Part $(a)$

$(a)$ Distance along $z$ -axis, $z = \frac{R}{\sqrt{2}} .$

Part $(b)$

$(b)$ Electrical field intensity at this time, $E = \frac{q}{4 蟺系_{o} R^{2}} 脳 \frac{2}{3 \sqrt{3}} .$

Step by step solution

01

Electrical field

When charge is present in whatever form, any point in space has an electrical property.

02

Analyzing diagram:

Horizontal component,

$E_{z} = 2 d E_{z} = 2 E \cos 胃$

$(\cos 胃 = \frac{z}{\sqrt{R^{2} + z^{2}}})$

The vertical components are equal and opposite and cancel each other.

$E = 2 脳 \frac{1}{4 蟺系_{o}} 脳 (\frac{q}{2 蟺 R}) 脳 \frac{d l 脳 z}{{(R^{2} + z^{2})}^{3 / 2}}$

Applying integration and substituting the value to the equation,

$E = \frac{z}{4 蟺系_{o}} 脳 \frac{q}{2 蟺 R} 脳 \frac{z}{{(R^{2} + z^{2})}^{3 / 2}} {鈭�}_{0}^{R} d l$

$E = \frac{q}{4 蟺系_{o}} 脳 \frac{z}{{(R^{2} + z^{2})}^{3 / 2}}$

03

Distance along axis (part a)

$E_{x} = m a x$ when $\frac{d E_{z}}{d z} = 0$ differenting $E_{n} .$

$E_{z} = \frac{q}{4 蟺系_{o}} 脳 \frac{z}{{(R^{2} + z^{2})}^{3 / 2}}$

Equating we get,

$\frac{d E_{z}}{d z} = \frac{q}{4 蟺系_{o}} [\frac{1}{{(R^{2} + z^{2})}^{3 / 2}} + \frac{z (- \frac{3}{2}) (2 z)}{{(R^{2} + z^{2})}^{5 / 2}}]$

$R^{2} + z^{2} - 3 z^{2} = 0$

The distance along $z -$ axis is,

$z = \frac{R}{\sqrt{2}} .$

04

Electrical strength (part b)

The equation as,

E = \frac{q}{4 蟺 系_{o}} 脳 \frac{(\frac{R}{\sqrt{2}})}{{(R^{2} + \frac{R^{2}}{2})}^{3 / 2}}

Equating we get,

$E = \frac{q}{4 蟺系_{o}} 脳 \frac{R}{\sqrt{2} 脳 R^{3} {(\frac{3}{2})}^{3 / 2}}$

The electric field intensity is,

$E = \frac{q}{4 蟺系_{o} R^{2}} 脳 \frac{2}{3 \sqrt{3}} .$

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Most popular questions from this chapter

The combustion of fossil fuels produces micron-sized particles of soot, one of the major components of air pollution. The terminal speeds of these particles are extremely small, so they remain suspended in air for very long periods of time. Furthermore, very small particles almost always acquire small amounts of charge from cosmic rays and various atmospheric effects, so their motion is influenced not only by gravity but also by the earth's weak electric field. Consider a small spherical particle of radius $r$ , density $蚁$ , and charge $q$ . A small sphere moving with speed v experiences a drag force $F_{drag} = 6 蟺畏 r v$ , where $畏$ is the viscosity of the air. (This differs from the drag force you learned in Chapter 6 because there we considered macroscopic rather than microscopic objects.)

a. A particle falling at its terminal speed $v_{term}$ is in equilibrium with no net force. Write Newton's first law for this particle falling in the presence of a downward electric field of strength $E$ , then solve to find an expression for $v_{term}$ .

b. Soot is primarily carbon, and carbon in the form of graphite has a density of $2200 kg / m^{3}$ . In the absence of an electric field, what is the terminal speed in $mm / s$ of a $1.0 - 渭 m$ -diameter graphite particle? The viscosity of air at $20 掳 C$ is $1.8 脳 10^{- 5} kg / ms$ .

c. The earth's electric field is typically (150 N/C , downward). In this field, what is the terminal speed in $mm / s$ of a $1.0$ $渭 m$ -diameter graphite particle that has acquired 250 extra electrons?

Rank in order, from largest to smallest, the electric field strengthsE1 to E4 at points $1 to 4$ in FIGURE Q23.3. Explain.

A small object is released at point $3$ in the center of the capacitor in FIGURE $Q 23.11$ . For each situation, does the object move to the right, to the left, or remain in place? If it moves, does it accelerate or move at constant speed?

a. A positive object is released from rest.

b. A neutral but polarizable object is released from rest.

c. A negative object is released from rest.

Show that the on-axis electric field of a ring of charge has the expected behavior when $z 鈮� R$ and when $z 鈮� R$ .

Rank in order, from largest to smallest, the electric field strengths $E_{1}$ to $E_{5}$ at the five points inFIGURE $Q 23.11$ . Explain.

See all solutions

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