Surface-charge of a uniformly polarized sphere

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In summary, the conversation discusses finding the electric field inside a cavity and determining the total charge on the inner surface. It is clarified that the proper way to find the charge is through integration, which results in a value of zero due to the positive and negative bound charges on the northern and southern hemispheres. Therefore, the total charge on the inner surface is not equal to P*cos(theta)*4*Pi*(R_0)^2.
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Niles
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Homework Statement


Hi all.

Please take a look at #2 in http://www.physics.utoronto.ca/~hori/Courses/P352F/hw6.pdf [Broken]

(this is not homework, I'm just training).

Ok, I can find the electric field inside the cavity y superposition and it is E = (-1/ 3*e_o)*P. I know that the bound surface charge is P*cos(theta), where theta is the usual spherical coordinate. My question is: Is the total charge on the inner surface P*cos(theta)*4*Pi*(R_0)^2?
 
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Where is [tex]\theta[/tex] in your last sentence? In order to find the total charge, you must integrate - not multiply - since the surface charge varies with respect to [tex]\theta[/tex]. The proper way to find the surface charge would be

[tex]Q = \int_S \sigma(\theta) \, da = \int_0^{2\pi} d\phi \int_0^{\pi} \cos \theta \sin \theta d\theta = 0[/tex]

It's zero, because on the northern hemisphere the bound charge is positive, which is compensated by the symmetrically negative bound charge on the southern hemisphere.

In conclusion, the answer is no.
 

1. What is surface charge of a uniformly polarized sphere?

The surface charge of a uniformly polarized sphere refers to the distribution of electric charge on the surface of the sphere due to its polarization. This arises when the electric dipole moment of the sphere is not evenly distributed, resulting in a net charge on the surface of the sphere.

2. How is the surface charge of a uniformly polarized sphere calculated?

The surface charge density of a uniformly polarized sphere can be calculated using the formula σ = P cos θ, where σ is the surface charge density, P is the dipole moment per unit volume, and θ is the angle between the direction of polarization and the surface normal. The total surface charge can then be calculated by integrating the surface charge density over the surface of the sphere.

3. What factors affect the surface charge of a uniformly polarized sphere?

The surface charge of a uniformly polarized sphere is affected by the magnitude and direction of the dipole moment, the radius of the sphere, and the dielectric constant of the material. Additionally, the orientation of the sphere relative to an external electric field can also affect the surface charge.

4. How does the surface charge of a uniformly polarized sphere impact its electric field?

The surface charge of a uniformly polarized sphere creates an electric field that is perpendicular to the surface of the sphere. This electric field is stronger at the poles of the sphere and weaker at the equator. The direction of the electric field can also be affected by the orientation of the sphere's dipole moment relative to the external electric field.

5. Can the surface charge of a uniformly polarized sphere be changed?

Yes, the surface charge of a uniformly polarized sphere can be changed by altering the dipole moment, the orientation of the sphere, or the external electric field. Additionally, the surface charge can also be affected by the dielectric properties of the material surrounding the sphere.

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