Chapter 23: Q. 45 (page 655)

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

Short Answer

Expert verified

Electrical field of a ring of charge, $E = \frac{q}{4 π ϵ_{o}} \times \frac{z}{{(R^{2} + z^{2})}^{3 / 2}}$

Step by step solution

Step: 1 Electrical field:

The electric field radiates outward in altogether directions from the purpose charge. Spherical equipotential surfaces are represented by the circles. A vector of the individual fields are often wont to calculate the electrical field from any amount of point charges.

Step: 2 Horizontal component of Electrical field:

The Horizontal component is

$E_{z} = 2 d E_{z} E_{z} = 2 E \cos θ$

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

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,

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

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

Step: 3 Evaluate diagram:

Due to electric field at $P$ in $d E$ .

$I f z < < R$ :

$E = \frac{q z}{4 π ϵ_{o} R^{3}}$ (At $z = 0$ and center of circle $E = 0$ )

If $z > > R$

$E = \frac{q}{4 π ϵ_{o}} \times \frac{z}{z^{3} {[1 + \frac{R^{2}}{z^{2}}]}^{3 / 2}}$

$E = \frac{q}{4 π ϵ_{o}} \times \frac{1}{z^{2}}$

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

Short Answer

Step by step solution

Step: 1 Electrical field:

Step: 2 Horizontal component of Electrical field:

Step: 3 Evaluate diagram:

One App. One Place for Learning.

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Show that the on-axis electric field of a ring of charge has the expected behavior when z≪Rand whenz≫R.

Short Answer

Step by step solution

Step: 1 Electrical field:

Step: 2 Horizontal component of Electrical field:

Step: 3 Evaluate diagram:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Fluids

Solid State Physics

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Linear Momentum

Study anywhere. Anytime. Across all devices.

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