What is the radius of the imaginary concentric sphere that divides the electrostatic field of a metal sphere of a radius 20 cm and change of `8muC` in two regions of identical energy?

Options

A. 30 cm
B. 40 cm
C. 60 cm
D. 80 cm

Answer

To find the radius of the imaginary concentric sphere that divides the electrostatic field of a metal sphere into two regions of identical energy, we can follow these steps:

### Step 1: Understand the Concept of Electrostatic Energy
The electrostatic energy (U) stored in the electric field of a charged sphere can be expressed as:
\[ U = \frac{1}{2} \int E^2 \, dV \]
For a uniformly charged sphere, the electric field (E) outside the sphere at a distance \( r \) from the center is given by:
\[ E = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant and \( Q \) is the charge on the sphere.

### Step 2: Calculate the Total Energy of the Sphere
The total energy stored in the electric field of the sphere can be calculated by integrating from the surface of the sphere (radius \( R \)) to infinity:
\[ U = \int_R^{\infty} \frac{1}{2} \left( \frac{kQ}{r^2} \right)^2 \cdot 4\pi r^2 \, dr \]
This simplifies to:
\[ U = 2\pi k^2 Q^2 \int_R^{\infty} \frac{1}{r^4} \, dr \]

### Step 3: Perform the Integration
The integral can be evaluated:
\[ \int_R^{\infty} \frac{1}{r^4} \, dr = \left[ -\frac{1}{3r^3} \right]_R^{\infty} = \frac{1}{3R^3} \]
Thus, the total energy becomes:
\[ U = \frac{2\pi k^2 Q^2}{3R^3} \]

### Step 4: Divide the Energy into Two Equal Parts
To find the radius \( r \) of the imaginary sphere that divides this energy into two equal parts, we set:
\[ U_1 = U_2 = \frac{U}{2} \]
The energy \( U_1 \) up to radius \( r \) is:
\[ U_1 = \frac{2\pi k^2 Q^2}{3r^3} \]
Setting \( U_1 = \frac{U}{2} \):
\[ \frac{2\pi k^2 Q^2}{3r^3} = \frac{1}{2} \cdot \frac{2\pi k^2 Q^2}{3R^3} \]

### Step 5: Solve for \( r \)
Cancelling terms and rearranging gives:
\[ \frac{1}{r^3} = \frac{1}{2R^3} \]
Thus:
\[ r^3 = 2R^3 \]
Taking the cube root:
\[ r = R \cdot 2^{1/3} \]
Given \( R = 20 \, \text{cm} \):
\[ r = 20 \cdot 2^{1/3} \]

### Step 6: Calculate the Numerical Value
Calculating \( 2^{1/3} \approx 1.2599 \):
\[ r \approx 20 \cdot 1.2599 \approx 25.198 \, \text{cm} \]

### Final Answer
The radius of the imaginary concentric sphere is approximately \( 25.2 \, \text{cm} \).

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