A uniformly charged spherical distribution is illustrated below. A spherical shell (blue) of radius b has volume charge density and the interior sphere (yellow) of radius a has volume charge density p, What is the electric field strength E outside the distribution for r>b? (hint: use Gauss's _aw.) P₁ b a P2

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### Electric Field Distribution of a Spherical Charge Distribution

A uniformly charged spherical distribution is illustrated below. A spherical shell (blue) of radius \( b \) has volume charge density \( \rho_1 \) and the interior sphere (yellow) of radius \( a \) has volume charge density \( \rho_2 \). What is the electric field strength \( E \) outside the distribution for \( r > b \)? (Hint: use Gauss’s Law.)

#### Diagram Explanation:

The diagram presents a cross-section of two spherical regions:
1. **Yellow Sphere (inner sphere)**: This sphere has a radius denoted as \( a \) and a volume charge density of \( \rho_2 \).
2. **Blue Shell (outer shell)**: This shell extends from radius \( a \) to radius \( b \), with a volume charge density \( \rho_1 \).

The black line passing through the spheres represents a radial distance \( r \) which is greater than the outer radius \( b \).

In order to solve for the electric field \( E \) outside the spherical distribution (i.e., for \( r > b \)), we can employ Gauss's Law, which relates the electric flux through a closed surface to the charge enclosed by that surface.

#### Gauss's Law Formula:
\[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]

Based on this principle, considering a spherical Gaussian surface of radius \( r \) (where \( r > b \)):
1. Calculate the total charge enclosed by the spherical distributions.
2. Use the symmetry of the problem to ascertain that the electric field \( \vec{E} \) will be radially symmetric and constant over the Gaussian surface.

To find the total charge \( Q_{\text{enc}} \):
\[ Q_{\text{enc}} = \rho_2 \cdot \frac{4}{3} \pi a^3 + \rho_1 \cdot \left( \frac{4}{3} \pi b^3 - \frac{4}{3} \pi a^3 \right) \]

Thereafter, using Gauss's Law:
\[ E \cdot 4 \pi r^2 = \frac{Q_{\text{enc}}}{\epsilon_0} \]

By solving this equation, we
Transcribed Image Text:### Electric Field Distribution of a Spherical Charge Distribution A uniformly charged spherical distribution is illustrated below. A spherical shell (blue) of radius \( b \) has volume charge density \( \rho_1 \) and the interior sphere (yellow) of radius \( a \) has volume charge density \( \rho_2 \). What is the electric field strength \( E \) outside the distribution for \( r > b \)? (Hint: use Gauss’s Law.) #### Diagram Explanation: The diagram presents a cross-section of two spherical regions: 1. **Yellow Sphere (inner sphere)**: This sphere has a radius denoted as \( a \) and a volume charge density of \( \rho_2 \). 2. **Blue Shell (outer shell)**: This shell extends from radius \( a \) to radius \( b \), with a volume charge density \( \rho_1 \). The black line passing through the spheres represents a radial distance \( r \) which is greater than the outer radius \( b \). In order to solve for the electric field \( E \) outside the spherical distribution (i.e., for \( r > b \)), we can employ Gauss's Law, which relates the electric flux through a closed surface to the charge enclosed by that surface. #### Gauss's Law Formula: \[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \] Based on this principle, considering a spherical Gaussian surface of radius \( r \) (where \( r > b \)): 1. Calculate the total charge enclosed by the spherical distributions. 2. Use the symmetry of the problem to ascertain that the electric field \( \vec{E} \) will be radially symmetric and constant over the Gaussian surface. To find the total charge \( Q_{\text{enc}} \): \[ Q_{\text{enc}} = \rho_2 \cdot \frac{4}{3} \pi a^3 + \rho_1 \cdot \left( \frac{4}{3} \pi b^3 - \frac{4}{3} \pi a^3 \right) \] Thereafter, using Gauss's Law: \[ E \cdot 4 \pi r^2 = \frac{Q_{\text{enc}}}{\epsilon_0} \] By solving this equation, we
Below is the transcribed text from the provided image, formatted as it would appear on an educational website:

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### Multiple Choice Question

**Instructions:** Choose the correct option from the following choices.

#### Options:

A. \(0\)

B. \(\frac{d^2(p_1 - p_2) + b^2p_1}{3e_0b^2}\)

C. \(\frac{d^2p_2}{3e_0b^2}\)

D. \(\frac{d^2(p_1 + p_2) + b^2p_1}{3e_0b^2}\)

E. \(\frac{d^2(p_1 - p_2) + b^2p_1}{3e_0b}\)

---

There are no graphs or diagrams included in this question. The question is written in a mathematical format and contains algebraic expressions. Make sure to carefully evaluate each option to find the correct solution.
Transcribed Image Text:Below is the transcribed text from the provided image, formatted as it would appear on an educational website: --- ### Multiple Choice Question **Instructions:** Choose the correct option from the following choices. #### Options: A. \(0\) B. \(\frac{d^2(p_1 - p_2) + b^2p_1}{3e_0b^2}\) C. \(\frac{d^2p_2}{3e_0b^2}\) D. \(\frac{d^2(p_1 + p_2) + b^2p_1}{3e_0b^2}\) E. \(\frac{d^2(p_1 - p_2) + b^2p_1}{3e_0b}\) --- There are no graphs or diagrams included in this question. The question is written in a mathematical format and contains algebraic expressions. Make sure to carefully evaluate each option to find the correct solution.
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