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A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization P(r) = f. where k is a constant and r is the distance from the center (Fig. 4.18). (There is no free charge in the problem.) Find the clectric field in all three regions by two different methods: (a) Locate all the bound charge, and use Gauss's law (Eq. 2.13) to calculate the field it produces. (b) Use Eq. 4.23 to find D, and then get E from Eq. 4.21. [Notice that the second method is much faster, and avoids any explicit reference to the bound charges.] Figure 4.18 For any closed surface, then, E- da = Qene. (2.13) D = €qE + P, (4.21) da = Q fen " (4.23)

A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization P(r) = f. where k is a constant and r is the distance from the center (Fig. 4.18). (There is no free charge in the problem.) Find the clectric field in all three regions by two different methods: (a) Locate all the bound charge, and use Gauss's law (Eq. 2.13) to calculate the field it produces. (b) Use Eq. 4.23 to find D, and then get E from Eq. 4.21. [Notice that the second method is much faster, and avoids any explicit reference to the bound charges.] Figure 4.18 For any closed surface, then, E- da = Qene. (2.13) D = €qE + P, (4.21) da = Q fen " (4.23)

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### Spherical Shell with Dielectric Material **Problem Statement:** A thick spherical shell (inner radius \( a \), outer radius \( b \)) is made of dielectric material with a "frozen-in" polarization given by: \[ \mathbf{P}(r) = \frac{k}{r} \, \hat{\mathbf{r}}, \] where \( k \) is a constant, and \( r \) is the distance from the center. **Objective:** Find the electric field in all three regions by two different methods: 1. **Locate all the bound charge and use Gauss’s law (Eq. 2.13) to calculate the field it produces.** 2. **Use Eq. 4.23 to find \( \mathbf{D} \), and then get \( \mathbf{E} \) from Eq. 4.21.** Notice that the second method is much faster and avoids any explicit reference to the bound charges. **Diagram Explanation:** - **Figure 4.18**: Illustrates a cross-section of the thick spherical shell. It shows the polarization vectors (\( \mathbf{P} \)) at various points. These vectors are directed radially outwards, represented by arrows. **Key Equations and Their Meanings:** - **Gauss's Law** (Equation 2.13): \[ \oint \mathbf{E} \cdot d\mathbf{a} = \frac{1}{\varepsilon_0} Q_{\text{enc}} \] This equation relates the electric flux through a closed surface to the charge enclosed by that surface. - **Electric Displacement Field** (\( \mathbf{D} \)) (Equation 4.21): \[ \mathbf{D} \equiv \varepsilon_0 \mathbf{E} + \mathbf{P} \] This relation connects the electric field \( \mathbf{E} \), the electric displacement field \( \mathbf{D} \), and the polarization \( \mathbf{P} \). - **Electric Displacement Integral** (Equation 4.23): \[ \oint \mathbf{D} \cdot d\mathbf{a} = Q_{\text{free}} \] This equation is used to calculate the flux of \( \mathbf{D} \) across a surface enclosing free
Transcribed Image Text:### Spherical Shell with Dielectric Material **Problem Statement:** A thick spherical shell (inner radius \( a \), outer radius \( b \)) is made of dielectric material with a "frozen-in" polarization given by: \[ \mathbf{P}(r) = \frac{k}{r} \, \hat{\mathbf{r}}, \] where \( k \) is a constant, and \( r \) is the distance from the center. **Objective:** Find the electric field in all three regions by two different methods: 1. **Locate all the bound charge and use Gauss’s law (Eq. 2.13) to calculate the field it produces.** 2. **Use Eq. 4.23 to find \( \mathbf{D} \), and then get \( \mathbf{E} \) from Eq. 4.21.** Notice that the second method is much faster and avoids any explicit reference to the bound charges. **Diagram Explanation:** - **Figure 4.18**: Illustrates a cross-section of the thick spherical shell. It shows the polarization vectors (\( \mathbf{P} \)) at various points. These vectors are directed radially outwards, represented by arrows. **Key Equations and Their Meanings:** - **Gauss's Law** (Equation 2.13): \[ \oint \mathbf{E} \cdot d\mathbf{a} = \frac{1}{\varepsilon_0} Q_{\text{enc}} \] This equation relates the electric flux through a closed surface to the charge enclosed by that surface. - **Electric Displacement Field** (\( \mathbf{D} \)) (Equation 4.21): \[ \mathbf{D} \equiv \varepsilon_0 \mathbf{E} + \mathbf{P} \] This relation connects the electric field \( \mathbf{E} \), the electric displacement field \( \mathbf{D} \), and the polarization \( \mathbf{P} \). - **Electric Displacement Integral** (Equation 4.23): \[ \oint \mathbf{D} \cdot d\mathbf{a} = Q_{\text{free}} \] This equation is used to calculate the flux of \( \mathbf{D} \) across a surface enclosing free
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