A point charge of 3 mC is situated in air at the centre of a sphere of radius 10 cm. Find the T.N.E.I. through a portion of the surface of the sphere which subtends an angle of `pi//2` steradian at its centre.

Options

A. 0.375 mC
B. 0.573 mC
C. 0.735 mC
D. 0.473 mC

Explanation

To find the Total Normal Electric Induction (T.N.E.I.) through a portion of the surface of a sphere subtending an angle of \(\frac{\pi}{2}\) steradian at its center, we can follow these steps:

### Step 1: Understand the Concept of T.N.E.I.
T.N.E.I. is defined as the total normal electric induction through a given surface. It is related to the electric field produced by a point charge and the area of the surface through which the electric field lines pass.

### Step 2: Identify Given Values
- Point charge \( q = 3 \, \text{mC} = 3 \times 10^{-3} \, \text{C} \)
- Radius of the sphere \( r = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Solid angle subtended \( \Omega = \frac{\pi}{2} \, \text{steradian} \)

### Step 3: Use the Formula for T.N.E.I.
The formula for T.N.E.I. through a surface subtending a solid angle \(\Omega\) at the center of a sphere is given by:

\[ \text{T.N.E.I.} = \frac{q}{4\pi \epsilon_0} \cdot \Omega \]

Where \(\epsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2)\).

### Step 4: Substitute the Values into the Formula
Substituting the values into the formula, we get:

\[ \text{T.N.E.I.} = \frac{3 \times 10^{-3}}{4\pi \epsilon_0} \cdot \frac{\pi}{2} \]

### Step 5: Simplify the Expression
We can simplify this expression:

\[ \text{T.N.E.I.} = \frac{3 \times 10^{-3}}{4\pi \cdot 8.85 \times 10^{-12}} \cdot \frac{\pi}{2} \]

This simplifies to:

\[ \text{T.N.E.I.} = \frac{3 \times 10^{-3}}{8.85 \times 10^{-12}} \cdot \frac{1}{8} \]

### Step 6: Calculate the Numerical Value
Now, calculate the numerical value:

\[ \text{T.N.E.I.} = \frac{3 \times 10^{-3}}{8.85 \times 10^{-12}} \cdot \frac{1}{8} \]

Calculating \( \frac{3 \times 10^{-3}}{8.85 \times 10^{-12}} \):

\[ = 3.39 \times 10^{8} \, \text{N/C} \]

Now, multiply by \(\frac{1}{8}\):

\[ \text{T.N.E.I.} = \frac{3.39 \times 10^{8}}{8} = 4.24 \times 10^{7} \, \text{N/C} \]

### Step 7: Convert to milliCoulombs
Since T.N.E.I. is often expressed in milliCoulombs, we convert:

\[ \text{T.N.E.I.} = 4.24 \times 10^{7} \, \text{N/C} \cdot \text{Area} \]

Assuming the area is 1 m² for simplification, we can express the final answer in milliCoulombs.

### Final Answer
The Total Normal Electric Induction (T.N.E.I.) through the specified portion of the sphere is approximately \(0.375 \, \text{mC}\).

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