Potential inside sphere with empty cavity

In summary, an insulating sphere with uniform charge density has a spherical cavity of half its radius cut out. The magnitude and direction of the electric field at points A and B can be found using Gauss' law or the equation for the electric field of a volume charge distribution. The potential at points A and B can be found by treating the sphere as two separate spheres and taking the difference of their fields and potentials. An algebraic solution for the electric field and potential in the space outside the larger sphere can also be written using the radius vector of point C as r = RC.
  • #1
bfusco
128
1

Homework Statement


An insulating sphere of radius R , centered at point A, has uniform chagre density ρ. A spherical cavity of radius R / 2 , centered at point C, is then cut out and left empty, see Fig.

(a) Find magnitude and direction of the electric field at points A and B.

(b) Find the potential at points A and B. Set V(r → ∞) = 0.

(c) Write down an algebraic solution (no integrals!) for E(r) and V (r) for the space outside the larger sphere, r > R. Choose r = 0 at point A, and the radius vector of point C as r = RC .

(question1) in the uploaded file


The Attempt at a Solution


(a)First I want to figure out the volume charge distribution, which I wanted to do by finding the the volume of the whole sphere minus the volume of the cavity.
[tex] \int_V \rho \cdot dVolume = \frac{4}{3}\pi \rho (R^3-\frac{R^3}{8})=\frac{7}{6}\pi R^3 \rho [/tex]

Then to get [itex] E_B [/itex] I wanted to use Gauss' law, but I am not sure how i would set that up because the E field isn't isotropic, so instead I am trying to use the equation for E-field of a volume charge distribution:
[tex] E(r)=k\int \frac{dq \hat{s}}{s^2} [/tex]
Where i just calculated dq. [itex] s^2=R [/itex], but I am not sure what [itex] \hat{s} [/itex] is
 

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  • #2
thinking about it though, I am pretty sure that what i got for dq is wrong, because i don't think what i did takes into account that the cavity isn't centered at the origin
 
  • #3
Treat it as one sphere minus another. Find the fields and potentials for each sphere and take the difference.
 

1. What is the significance of a potential inside a sphere with an empty cavity?

The potential inside a sphere with an empty cavity is a measure of the electric potential at any point inside the sphere. It is affected by the charge distribution and the geometry of the sphere and its cavity. This potential can be used to calculate the electric field and other properties within the sphere.

2. How is the potential inside a sphere with an empty cavity calculated?

The potential inside a sphere with an empty cavity is calculated using the Poisson or Laplace equations, which relate the potential to the charge distribution and the geometry of the sphere. These equations can be solved analytically or numerically depending on the specific case.

3. What is the relationship between the potential inside a sphere and the potential at its surface?

The potential inside a sphere with an empty cavity is directly related to the potential at its surface. In fact, the potential inside the sphere is equal to the potential at the surface if the sphere is conducting. This relationship is important for understanding the behavior of electric fields and charges within the sphere.

4. How does the presence of a charge affect the potential inside a sphere with an empty cavity?

The presence of a charge inside the sphere or on its surface will affect the potential inside the sphere. The distribution of the charge and its magnitude will determine the specific potential values and how it varies within the sphere. This is an important factor to consider in many applications, such as in electronic devices.

5. Can the potential inside a sphere with an empty cavity ever be zero?

Yes, the potential inside a sphere with an empty cavity can be zero at certain points within the sphere. This occurs when the electric field at that point is equal and opposite to the electric field caused by any charges inside or outside the sphere. This can happen, for example, at the center of the sphere or at points where the electric field is canceled out due to symmetry.

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