The intensity of an electric field depends only on the co-ordinates x, y and z as follows:
`vecE=a((xhati+yhatj+zhatk))/((x^2+y^2+z^2)^(3//2))`unit.
The electrostatic energy stored between two imaginary concentric spherical shells of radii R and 2R with centre at origin is

Options

A. `(4piepsilon_0a^2)/R`
B. `(2piepsilon_0a^2)/R`
C. `(piepsilon_0a^2)/R`
D. `(piepsilon_0a^2)/(2R)`

Answer

To find the electrostatic energy stored between two concentric spherical shells of radii \( R \) and \( 2R \), we can follow these steps:

### Step 1: Understand the electric field
The electric field is given by:
\[ \vec{E} = \frac{a (x \hat{i} + y \hat{j} + z \hat{k})}{(x^2 + y^2 + z^2)^{3/2}} \]
This can be expressed in terms of the position vector \( \vec{r} \):
\[ \vec{E} = \frac{a \vec{r}}{|\vec{r}|^3} \]
where \( |\vec{r}| = \sqrt{x^2 + y^2 + z^2} \).

### Step 2: Calculate the energy density
The energy density \( u \) in an electric field is given by:
\[ u = \frac{1}{2} \epsilon_0 E^2 \]
Substituting the expression for \( E \):
\[ E = \frac{a}{r^2} \quad \text{(since } |\vec{r}| = r\text{)} \]
Thus,
\[ E^2 = \left(\frac{a}{r^2}\right)^2 = \frac{a^2}{r^4} \]
Now, substituting this into the energy density formula:
\[ u = \frac{1}{2} \epsilon_0 \frac{a^2}{r^4} \]

### Step 3: Calculate the differential volume element
The differential volume element \( dv \) for a spherical shell is given by:
\[ dv = 4\pi r^2 dr \]

### Step 4: Write the expression for differential energy
The differential energy \( du \) stored in the shell can be expressed as:
\[ du = u \cdot dv = \frac{1}{2} \epsilon_0 \frac{a^2}{r^4} \cdot 4\pi r^2 dr \]
This simplifies to:
\[ du = 2\pi \epsilon_0 a^2 \frac{1}{r^2} dr \]

### Step 5: Integrate to find total energy
To find the total energy \( U \) stored between the shells from \( r = R \) to \( r = 2R \), we integrate \( du \):
\[ U = \int_{R}^{2R} du = \int_{R}^{2R} 2\pi \epsilon_0 a^2 \frac{1}{r^2} dr \]
Calculating the integral:
\[ U = 2\pi \epsilon_0 a^2 \left[-\frac{1}{r}\right]_{R}^{2R} = 2\pi \epsilon_0 a^2 \left(-\frac{1}{2R} + \frac{1}{R}\right) \]
This simplifies to:
\[ U = 2\pi \epsilon_0 a^2 \left(\frac{1}{R} - \frac{1}{2R}\right) = 2\pi \epsilon_0 a^2 \left(\frac{1}{2R}\right) = \frac{\pi \epsilon_0 a^2}{R} \]

### Final Result
Thus, the total electrostatic energy stored between the two spherical shells is:
\[ U = \frac{\pi \epsilon_0 a^2}{R} \]

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