Q. 45 Show that the on-axis electric f... [FREE SOLUTION]

Chapter 23: Q. 45 (page 655)

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

Short Answer

Expert verified

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

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

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

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

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,

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

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

Step: 3 Evaluate diagram:

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

$I f z < < R$ :

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

If $z > > R$

$E = \frac{q}{4 蟺系_{o}} 脳 \frac{z}{z^{3} {[1 + \frac{R^{2}}{z^{2}}]}^{3 / 2}}$

$E = \frac{q}{4 蟺系_{o}} 脳 \frac{1}{z^{2}}$

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91影视

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

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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91影视

Show that the on-axis electric field of a ring of charge has the expected behavior when z鈮�Rand whenz鈮�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

Waves Physics

Electricity

Turning Points in Physics

Atoms and Radioactivity

Magnetism and Electromagnetic Induction

Kinematics Physics

Study anywhere. Anytime. Across all devices.

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