Find the electrostatic potential due to a hoop of charge

In summary, the problem involves calculating the electrostatic potential at all points on the x axis due to a hoop of charge of radius k. The result is expected to be (2*pi*j)/|x| far from the hoop on the x axis. The leading behaviour of the potential far from the hoop is expected to decrease inverse proportionally to the distance from the hoop. This leads to the limiting forms of the equipotential surfaces and field lines being perpendicular to the x axis and converging towards the hoop as the distance increases.
  • #1
wam_mi
81
1

Homework Statement



(Part 1) A hoop of charge of radius k lies in the y, z plane, centred on the x axis, so that
it occupies the points (0, y, z) with y^2 + z^2 = k^2. If the (linear) charge density
in the hoop is j, calculate the electrostatic potential Fi at all points on the
x axis, and show that, far from the hoop on the x axis, Fi (x, 0, 0) = (2*pi*j)/|x|.
Explain briefly why this result was only to be expected.

(Part 2) More generally, what do you expect the leading behaviour of Fi(r) to be, far
from the hoop? (You do not need to give any detailed calculations.) Use your
answer to deduce the limiting forms of the equipotential surfaces and field lines,
again far from the hoop.


Homework Equations




The Attempt at a Solution



(Part 1)

Total Charge Q = Integrating j * dl with interval [0 , 2*pi]
= 2*pi*k*j

By Symmetry, Fi (x) = 2*pi*k*j / |x| (Is this wrong?)

And now I'm stuck for the rest of the question.

Please help! Thank you!
 
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  • #2
Integrating the charge elements and using symmetry is involved for solving this problem.

[tex]E = E_x =\int\mbox{dEcos}\theta\mbox{d}\theta[/tex]

where

[tex]dE =\frac{Kd\rho}{r^2}[/tex]

The sine components of E cancel due to symmetry, and

[tex]cos\theta = \frac{x}{\sqrt{k^2+x^2}}[/tex]

The charge element is

[tex]d\rho = jkd\phi \ \mbox{and}\ d\phi \ \mbox{ is integrated from 0 to} \ 2\pi[/tex]

Once E is found, the potential is found from integrating E from infinity to x along the x axis.
 

Related to Find the electrostatic potential due to a hoop of charge

1. What is electrostatic potential?

Electrostatic potential is the amount of work required to move a unit positive charge from infinity to a specific point in an electric field.

2. How is the electrostatic potential of a charged hoop calculated?

The electrostatic potential due to a hoop of charge can be calculated using the formula V = kQ/r, where V is the potential, k is the Coulomb's constant, Q is the charge of the hoop, and r is the distance from the center of the hoop to the point where the potential is being measured.

3. What is the direction of the electrostatic potential for a charged hoop?

The direction of the electrostatic potential for a charged hoop is radial, meaning it points away from the center of the hoop towards the surrounding environment.

4. Can the electrostatic potential of a charged hoop be negative?

Yes, the electrostatic potential can be negative. This would occur if the charge of the hoop is negative or if the point where the potential is being measured is closer to the hoop than its center.

5. How does the electrostatic potential of a charged hoop vary with distance from the center?

The electrostatic potential of a charged hoop follows an inverse relationship with distance from the center. This means that as the distance increases, the potential decreases, and vice versa.

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