Question

The intensity of an electric field depends only on the coordinates x, y, and z as follows:

E = a [(xi + yj + zk)]/[(x^2 + y^2 + z^2)]^3/2

The electrostatic energy stored between 2 imaginary concentric spherical shells of radii R and 2R with center at the origin is?

[Ans: (pi)ε0 a^2/R ]

Answer 1

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Answer 2

The electrostatic energy stored is $\bold{\frac{\pi \epsilon_0 a^{2}}{R}}$.

Solution:

Now, the intensity of the electric field becomes,

$\bold{E =\frac{a(xi + yj + zk)}{(x^2 + y^2 + z^2)^{3/2}}}$

Let's us consider that,

$\bold{xi + yj + zk = r}$

$\bold{x^2 + y^2 + z^2 = r^2}$

Now, the electric field becomes,

$\bold{E= \frac{ar}{(r^2)^{3/2}}}$

On squaring on both the sides, we get,

$\bold{E^{2}=\frac{a^{2} r^{2}}{r^{6}}}$

$\bold{E^2=\frac{a^{2}}{r^{4}}}$

Now, the electrostatic energy is given by the formula:

$\bold{\int_{0}^{v} d U=\int_{R}^{2R} \frac{1}{2} \epsilon_0 E^{2} 4 \pi r^{2} d r}$

$\bold{U=\frac{1}{2} \epsilon_0 \int_{R}^{2 R} \frac{a^{2}}{r^{4}} 4 \pi r^{2} d r}$

$\bold{U=2\pi\epsilon_0a^2 \int_R^{2R} \frac{dr}{r^{2}}}$

Now, on integrating, we get,

$\bold{U=2 \pi \epsilon_0 a^{2} \times \left(\frac{-1}{r}\right)_{R}^{2 R}}$

On substituting the limits, we get,

$\bold{U=\frac{2 \pi \epsilon_0 a^{2}}{2 R}}$

$\bold{\therefore U=\frac{\pi \epsilon_0 a^{2}}{R}}$