Gauss' law for a cavity in an insulator

In summary, Gauss' law for a cavity in an insulator states that the electric field inside a cavity within an insulating material depends solely on the charge enclosed within the cavity. The law implies that any external charges or fields do not influence the electric field inside the cavity, as the insulating material prevents the movement of charges. Consequently, the total electric flux through a closed surface surrounding the cavity is proportional to the enclosed charge, following the principle that only the charge inside the cavity contributes to the electric field within it.
  • #1
laser
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Homework Statement
A solid sphere of radius R has uniform charge density ρ. A hole of radius R/2 is scooped out of it as shown in Figure 10. Show that the field inside the hole is uniform and along the x-axis and of magnitude ρR/6ε0. Hint: Think of the hole as a superposition of positive and negative charges.
Relevant Equations
E=kq/r^2
This is a problem from Yale OCW (Shankar). The solution he gives is as follows:

Screenshot_3.png


Sure, this makes sense. However...

Superimposed rho and negative rho with radius R/2 means there is no charge enclosed in the cavity... therefore
no charge -> no flux -> no electric field.
 
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  • #2
laser said:
Homework Statement: A solid sphere of radius R has uniform charge density ρ. A hole of radius R/2 is scooped out of it as shown in Figure 10. Show that the field inside the hole is uniform and along the x-axis and of magnitude ρR/6ε0. Hint: Think of the hole as a superposition of positive and negative charges.
Relevant Equations: E=kq/r^2

This is a problem from Yale OCW (Shankar). The solution he gives is as follows:

View attachment 340207

Sure, this makes sense. However...

Superimposed rho and negative rho with radius R/2 means there is no charge enclosed in the cavity... therefore
no charge -> no flux -> no electric field.
No. You cannot draw the conclusion that there is no electric field. You would need to be able to make some symmetry argument for that to hold.

No net flux only means any field lines that come in also go out somewhere else.
 
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FAQ: Gauss' law for a cavity in an insulator

What is Gauss' Law for a cavity in an insulator?

Gauss' Law for a cavity in an insulator states that the electric flux through a closed surface surrounding a cavity within an insulating material is proportional to the net charge enclosed within that cavity. Mathematically, it is expressed as ∮E·dA = Q_enc/ε₀, where E is the electric field, dA is a differential area on the closed surface, Q_enc is the net charge enclosed within the cavity, and ε₀ is the permittivity of free space.

How does the presence of a cavity affect the electric field in an insulator?

The presence of a cavity in an insulator affects the electric field such that if the cavity contains no charge, the electric field inside the cavity is zero. If there is a charge inside the cavity, the electric field inside the cavity is determined by the charge distribution according to Gauss' Law, while the field in the insulating material outside the cavity remains unaffected by the charge inside the cavity.

Can Gauss' Law be applied to a cavity in a conductor?

Gauss' Law can indeed be applied to a cavity in a conductor, but the results differ from those in an insulator. In a conductor, any excess charge resides on the surface, and the electric field inside a conductor in electrostatic equilibrium is zero. If there is a charge inside a cavity within a conductor, it induces an equal and opposite charge on the inner surface of the cavity, maintaining zero electric field within the conducting material itself.

What happens if the cavity in the insulator contains a charge?

If the cavity in the insulator contains a charge, the electric field inside the cavity is non-zero and is determined by the charge distribution within the cavity. The electric field outside the cavity in the insulating material remains unaffected by the presence of the charge within the cavity, as the insulating material does not allow free movement of charges to neutralize the field.

How does the shape of the cavity influence the application of Gauss' Law?

The shape of the cavity does not influence the fundamental application of Gauss' Law, which depends only on the net charge enclosed within a closed surface. However, the symmetry of the cavity can simplify calculations. For highly symmetric shapes like spherical or cylindrical cavities, the electric field can be more easily determined using Gauss' Law due to the uniform distribution of the field lines.

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