Gauss' Law with Calculus Video Lessons

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Problem: An insulating sphere of radius a, centered at the origin, has a uniform volume charge density  ρ.A spherical cavity is excised from the inside of the sphere. The cavity has radius  a/4 and is centered at position  h, where |h| < 3a/4 , so that the entire cavity is contained within the larger sphere. Find the electric field  inside the cavity.Express your answer as a vector in terms of any or all of  ρ, ε0, r, and h.

FREE Expert Solution

Gauss' law:

E·dA=qencε0E·A=qε0

The cavity is inside the sphere, the sphere is charged, and the electric field is to be found inside the cavity.

A Gaussian surface completely inside the cavity encloses no charge.

E·dA=0ε0E·A=0

Charge in a gaussian surface,

Area of a sphere, A:

A=4πr2

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Problem Details

An insulating sphere of radius a, centered at the origin, has a uniform volume charge density  ρ.

A spherical cavity is excised from the inside of the sphere. The cavity has radius  a/4 and is centered at position  h, where |h| < 3a/4 , so that the entire cavity is contained within the larger sphere. Find the electric field  inside the cavity.

Express your answer as a vector in terms of any or all of  ρ, ε0, r, and h.

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