Suppose the field inside a large piece of dielectric is E0, so that the electric displacement is D0=
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The electric displacement is given by the equation: \[ D = \varepsilon E \] where \( \varepsilon \) is the permittivity of the dielectric. Show more…
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The electric field $\vec{E}_{0}$ in a spherical cavity in a uniform dielectric of permittivity $\varepsilon$ is related to the far away field $\vec{E}$, in the following manner. Imagine the cavity to be filled up with the dielectric. Then there will be a uniform field $E$ everywhere and a polarization $\overrightarrow{P,}$ given by, $\vec{P}=(\varepsilon-1) \varepsilon_{0} \vec{E}$ Now take out the sphere making the cavity, the electric field inside the sphere will be $-\frac{P}{3 \varepsilon_{0}}$ By superposition. $\overrightarrow{E_{0}}-\frac{\vec{P}}{3 \varepsilon_{0}}=\vec{E}$ or, $\overrightarrow{E_{0}}=\vec{E}+\frac{1}{3}(\varepsilon-1) \vec{E}=\frac{1}{3}(\varepsilon+2) \vec{E}$
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Let the field in the dielectric be $\vec{E}$ making an angle $\alpha$ with $\vec{n} .$ Then we have the boundary conditions, $E_{0} \cos \alpha_{0}=\varepsilon E \cos \alpha$ and $E_{0} \sin c_{0}=E \sin \alpha$ So $E=E_{0} \sqrt{\sin ^{2} \alpha_{0}+\frac{1}{\varepsilon^{2}} \cos ^{2} \alpha_{0}}$ and $\tan \alpha=\varepsilon \tan \alpha_{0}$ In the dielectric the normal component of the induction vector is $D_{n}=\varepsilon_{0} \varepsilon E_{n}=\varepsilon_{0} \varepsilon E \cos \alpha=\varepsilon_{0} E_{0} \cos \alpha_{0}$ $\sigma^{\prime}=P_{n}=D_{n}-\varepsilon_{0} E_{n}=\left(1-\frac{1}{\varepsilon}\right) \varepsilon_{0} E_{0} \cos \alpha_{0}$ or, $\quad \sigma^{\prime}=\frac{\varepsilon-1}{\varepsilon} \varepsilon_{0} E_{0} \cos \alpha_{0}$
The volume between two concentric conducting spheres of radii a and b (a < b) is filled with an inhomogeneous dielectric constant where A is a constant and r is the radial component. ε = ε0 / (1 + Ar) A charge Q is placed on the inner surface while the outer surface is grounded. Find: - The electric displacement in the region between the spheres - The bound charge density in the region between the spheres - The bound surface charge density at r = a and r = b.
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