Potential of infinite sheet with thickness

However, there is an easier way to calculate the potential outside the sheet using the electric field expression you already have. Since the electric field is constant outside the sheet, you can use the formula V = -Ed to find the potential at any point outside the sheet. In summary, the potential inside an infinite insulating sheet with uniform density ρ and thickness d can be calculated as V_f = -ρx^2/ε_0 for x < d/2, with V_i = 0 at x = 0. Outside the sheet, the potential can be found using the formula V = -Ed, where the electric field is given by E = ρd/ε_0 and is constant everywhere outside the sheet.
  • #1
susdu
24
0

Homework Statement



Describe the potential inside and outside an infinite insulating sheet with uniform density ρ and thickness d, as a function of x (distance from the center of the sheet). zero potential has been set at its center. What is the potential on the surface of the sheet?

Homework Equations



Potential and E.field definitions.

The Attempt at a Solution



I know that inside the sheet ([itex] x<\frac{d}{2}[/itex]) the field is given by

[itex] E=\frac{ρx}{ε_0}[/itex].

so, potential inside the sheet (with ##V_i=0## at ## x=0##) is:

##V_f=-\int^x_0 E\,ds=-\frac{ρx^2}{ε_0}##

similarly, outside the sheet ([itex] x>\frac{d}{2}[/itex]) field is

[itex] E=\frac{ρd}{ε_0}[/itex].

however, I'm not sure about the expression for outside the sheet, is it

##V_f=-\int^\frac{d}{2}_0 E\,ds-\int^x_\frac{d}{2} E\,ds##

?
 
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  • #2
I would use the Gauss Law. Let's consider a cylindrical surface with it's axis perpendicular to the sheet that "sticks out" of the sheet equally. Gauss Law states, that:

[tex]\epsilon _0 E \int ds = Q[/tex]

And Q=density of charge x volume.
From symmetry we know that there's no flux through the curved surface of the cyllinder, so we have flux only through bases of the cyllinder:

[tex]\epsilon _0 E 2\pi R^2 = \rho \pi R^2 d[/tex]

So [tex]E=\frac{\rho d}{2\epsilon _0}[/tex]

And E is constant everywhere outside the sheet.
 
  • #3
I already know the expressions for E.field outside/inside the sheet.
I need an expression for potential.
 
  • #4
susdu said:
##V_f=-\int^\frac{d}{2}_0 E\,ds-\int^x_\frac{d}{2} E\,ds##?
Looks right to me.
 
  • #5


I would like to point out that the expression for the potential outside the sheet is not correct. The correct expression for the potential outside the sheet is:

##V_f=-\int^\frac{d}{2}_0 E\,ds-\int^x_\frac{d}{2} E\,ds+V_s##

Where ##V_s## is the potential at the surface of the sheet, which can be calculated using the boundary conditions. For an infinite insulating sheet, the potential at the surface is equal to the potential at infinity, which is usually taken as zero. Therefore, the potential outside the sheet can be written as:

##V_f=-\int^\frac{d}{2}_0 E\,ds-\int^x_\frac{d}{2} E\,ds+0##

This expression can be simplified to:

##V_f=-\frac{ρd^2}{2ε_0}-\frac{ρ(x-\frac{d}{2})^2}{2ε_0}##

Which can also be written as:

##V_f=-\frac{ρ}{2ε_0}(d^2+(x-\frac{d}{2})^2)##

This expression shows that the potential outside the sheet decreases with distance from the center of the sheet, and it is at its lowest value at the surface of the sheet.

I would also like to mention that the potential inside the sheet can be written as:

##V_f=-\frac{ρ}{2ε_0}(d^2-x^2)##

This expression shows that the potential inside the sheet increases with distance from the center of the sheet, and it is at its highest value at the center of the sheet.

In conclusion, the potential inside and outside an infinite insulating sheet with uniform density and thickness can be calculated using the expressions mentioned above. The potential on the surface of the sheet is equal to the potential at infinity, which is usually taken as zero.
 

1. What is an infinite sheet with thickness?

An infinite sheet with thickness is a hypothetical object used in mathematical and scientific calculations. It is a two-dimensional plane that extends infinitely in all directions, but has a finite thickness. This means that it has an infinite surface area, but a finite volume.

2. What is the potential of an infinite sheet with thickness?

The potential of an infinite sheet with thickness is the measure of the electric potential energy per unit charge at any point in space surrounding the sheet. It is a scalar quantity and is typically denoted by the symbol V. The potential decreases as the distance from the sheet increases.

3. How is the potential of an infinite sheet with thickness calculated?

The potential of an infinite sheet with thickness is calculated using the formula V = σ/2ε, where V is the potential, σ is the surface charge density of the sheet, and ε is the permittivity of the surrounding medium. This formula assumes that the sheet has a uniform charge distribution.

4. What is the significance of an infinite sheet with thickness in physics?

An infinite sheet with thickness is a useful concept in physics because it allows us to simplify complex calculations involving electric fields and potentials. It is often used as a model for objects with large surface areas, such as parallel plate capacitors or large conducting plates.

5. Are there any real-life examples of an infinite sheet with thickness?

No, an infinite sheet with thickness is a purely theoretical concept and does not exist in the physical world. However, it can be used as an approximation for objects with very large surface areas, such as the Earth's surface, which can be considered an infinite sheet with thickness compared to the size of a single point on its surface.

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