Show That A nonconducting sphere has a uniform volume charge...

Show That A nonconducting sphere has a uniform volume charge density $\rho$. Let $\vec{r}$ be the vector from the center of the sphere to a general point $P$ within the sphere. (a) Show that the electric field at $P$ is given by $\vec{E}=\rho \vec{r}$ $13 \varepsilon_{0}$. (Note that the result is independent of the radius of the sphere.) (b) A spherical cavity is hollowed out of the sphere, as shown in Fig. 24-30. Using superposition concepts, show that the electric field at all points within the cavity is uniform and equal to $\bar{E}=\rho \vec{a} / 3 \varepsilon_{0}$, where $\vec{a}$ is the position vector from the center of the sphere to the center of the cavity. (Note that this result is independent of the radius of the sphere and the radius of the cavity.)

Show That A nonconducting sphere has a uniform volume charge density $\rho$. Let $\vec{r}$ be the vector from the center of the sphere to a general point $P$ within the sphere. (a) Show that the electric field at $P$ is given by $\vec{E}=\rho \vec{r}$ $13 \varepsilon_{0}$. (Note that the result is independent of the radius of the sphere.) (b) A spherical cavity is hollowed out of the sphere, as shown in Fig. 24-30. Using superposition concepts, show that the electric field at all points within
the cavity is uniform and equal to $\bar{E}=\rho \vec{a} / 3 \varepsilon_{0}$, where $\vec{a}$ is the position vector from the center of the sphere to the center of the cavity. (Note that this result is independent of the radius of the sphere and the radius of the cavity.)

Key Concepts

Superposition Principle

The superposition principle states that the total electric field created by multiple charge distributions is the vector sum of the fields produced by each distribution individually. This concept is crucial for determining the field in a cavity inside a charged object, as it allows for the treatment of the cavity as an additional charge distribution with an opposite sign to the original charge distribution, leading to a uniform electric field inside the cavity.

Electric Field in a Uniformly Charged Sphere

For a sphere with a uniform volume charge density, the electric field at a point inside the sphere increases linearly with distance from the center. This is derived using Gauss's Law, which shows that the field is proportional to the enclosed charge and, for a uniformly charged sphere, the enclosed charge scales with the volume of a sphere of radius equal to the distance from the center.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is especially useful in cases with high symmetry, such as spherical symmetry, allowing for straightforward calculation of the electric field both inside and outside charged objects.

Uniform Volume Charge Density

A uniform volume charge density means that the charge per unit volume is constant throughout the region of interest. This simplifies calculations since the total charge can be readily determined by multiplying the charge density by the volume, and it leads to relations where the electric field is directly proportional to the distance from the center for a spherically symmetric distribution.

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