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, $E = \frac{q}{4 π ϵ_{o} R^{2}} \times \frac{2}{3 \sqrt{3}} .$

Step by step solution

Electrical field

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

Analyzing diagram:

Horizontal component,

$E_{z} = 2 d E_{z} = 2 E \cos θ$

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

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

$E = 2 \times \frac{1}{4 π ϵ_{o}} \times (\frac{q}{2 π R}) \times \frac{d l \times z}{{(R^{2} + z^{2})}^{3 / 2}}$

Applying integration and substituting the value to the equation,

$E = \frac{z}{4 π ϵ_{o}} \times \frac{q}{2 π R} \times \frac{z}{{(R^{2} + z^{2})}^{3 / 2}} \int_{0}^{R} d l$

$E = \frac{q}{4 π ϵ_{o}} \times \frac{z}{{(R^{2} + z^{2})}^{3 / 2}}$

Distance along axis (part a)

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

$E_{z} = \frac{q}{4 π ϵ_{o}} \times \frac{z}{{(R^{2} + z^{2})}^{3 / 2}}$

Equating we get,

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

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

The distance along $z -$ axis is,

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

Electrical strength (part b)

The equation as,

E = \frac{q}{4 π ϵ_{o}} \times \frac{(\frac{R}{\sqrt{2}})}{{(R^{2} + \frac{R^{2}}{2})}^{3 / 2}}

Equating we get,

$E = \frac{q}{4 π ϵ_{o}} \times \frac{R}{\sqrt{2} \times R^{3} {(\frac{3}{2})}^{3 / 2}}$

The electric field intensity is,

$E = \frac{q}{4 π ϵ_{o} R^{2}} \times \frac{2}{3 \sqrt{3}} .$

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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

Step by step solution

Electrical field

Analyzing diagram:

Distance along axis (part a)

Electrical strength (part b)

One App. One Place for Learning.

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A ring of radius Rhas total chargeQ.aAt what distance along the z-axis is the electric field strength a maximum? bWhat is the electric field strength at this point?

Short Answer

Step by step solution

Electrical field

Analyzing diagram:

Distance along axis (part a)

Electrical strength (part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Fluids

Mechanics and Materials

Physical Quantities And Units

Wave Optics

Force

Study anywhere. Anytime. Across all devices.

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?