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Electric Field Inside A Cavity In An Insulating Sphere

Electric field inside a cavity in an insulating sphere :- Consider a uniformly charged insulating sphere with a volume charge density ρ C/m³. Inside the sphere, there is a spherical cavity whose center is at point C_₁. We need to find the electric field intensity inside the cavity at point P. Here, $\displaystyle \overrightarrow{x}$ and $\displaystyle \overrightarrow{y}$ are the position vectors of point P from the center of the sphere (C) and from the center of the cavity (C_₁), respectively.

If there were no cavity, the electric field at point P would be given by :

$\displaystyle {{{\vec{E}}}_{1}}={{{\vec{E}}}_{inside}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{Q}{{{R}^{3}}}\overrightarrow{x}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{\left( \frac{4}{3}\pi {{R}^{3}}\rho \right)}{{{R}^{3}}}\overrightarrow{x}=\frac{\rho }{3{{\varepsilon }_{0}}}\overrightarrow{x}$

Similarly, if we assume that the charge were only in the cavity, then it would form a uniformly charged sphere, and the electric field at point P would be :

$\displaystyle {{{\vec{E}}}_{2}}=\frac{\rho }{3{{\varepsilon }_{0}}}\overrightarrow{y}$

But since there is no charge in the cavity, the resulting electric field at point P is given by :

$\displaystyle {{{\vec{E}}}_{net}}={{{\vec{E}}}_{1}}-{{{\vec{E}}}_{2}}$

$\displaystyle \Rightarrow {{{\vec{E}}}_{net}}=\frac{\rho }{3{{\varepsilon }_{0}}}\overrightarrow{x}-\frac{\rho }{3{{\varepsilon }_{0}}}\overrightarrow{y}$

$\displaystyle \Rightarrow {{{\vec{E}}}_{net}}=\frac{\rho }{3{{\varepsilon }_{0}}}\left[ \overrightarrow{x}-\overrightarrow{y} \right]$

In the above figure :

$\displaystyle \overrightarrow{a}+\overrightarrow{y}=\overrightarrow{x}$

$\displaystyle \Rightarrow \left( \overrightarrow{x}-\overrightarrow{y} \right)=\overrightarrow{a}$

Hence Electric Field Inside A Cavity In An Insulating Sphere is given by :

$\displaystyle {{{\vec{E}}}_{net}}=\frac{\rho }{3{{\varepsilon }_{0}}}\overrightarrow{a}$ …..(1)

Equation (1) shows that the electric field inside the cavity is the same at all points and is directed along the line joining the center of the sphere (C) and the center of the cavity (C_₁).

Similarly, we can find the electric field intensity inside a cylindrical cavity within a long, uniformly charged insulating cylinder with volume charge density ρ C/m³.

If the distance between the axis of the cylindrical cavity and the axis of the cylinder is “a”, then, based on the above analysis of a cavity in an insulating sphere, the electric field intensity inside a cylindrical cavity is given by :

$\displaystyle \vec{E}=\frac{\rho }{2{{\varepsilon }_{0}}}\overrightarrow{a}$

Electric Field Inside A Cavity In An Insulating Sphere

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