Two small spherical shells a and B are given positive charges of 9 C and 4 C respectively and placed such that their centres are separated by 10 m. If P is a point in between them, where the electric field intensity is zero, then the distance of the point P from the centre of A is

Options

A. 5 m
B. 6 m
C. 7m
D. 8 m

Answer

To solve the problem of finding the distance of point P from the center of spherical shell A where the electric field intensity is zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have two spherical shells, A and B, with charges \( Q_A = 9 \, C \) and \( Q_B = 4 \, C \) respectively. The distance between their centers is \( 10 \, m \).

2. **Defining Variables**:
Let the distance from the center of shell A to point P be \( r \). Consequently, the distance from the center of shell B to point P will be \( 10 - r \).

3. **Electric Field Due to a Charged Shell**:
The electric field \( E \) due to a charged shell at a distance \( r \) from its center is given by:
\[ E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \]
where \( Q \) is the charge of the shell and \( \epsilon_0 \) is the permittivity of free space.

4. **Setting Up the Equation**:
At point P, the electric field due to shell A must equal the electric field due to shell B for the net electric field to be zero:
\[ E_A = E_B \]
This translates to:
\[ \frac{1}{4 \pi \epsilon_0} \frac{9}{r^2} = \frac{1}{4 \pi \epsilon_0} \frac{4}{(10 - r)^2} \]
We can cancel \( \frac{1}{4 \pi \epsilon_0} \) from both sides:
\[ \frac{9}{r^2} = \frac{4}{(10 - r)^2} \]

5. **Cross Multiplying**:
Cross multiplying gives us:
\[ 9(10 - r)^2 = 4r^2 \]

6. **Expanding the Equation**:
Expanding the left side:
\[ 9(100 - 20r + r^2) = 4r^2 \]
This simplifies to:
\[ 900 - 180r + 9r^2 = 4r^2 \]

7. **Rearranging the Equation**:
Rearranging the equation leads to:
\[ 9r^2 - 4r^2 - 180r + 900 = 0 \]
Which simplifies to:
\[ 5r^2 - 180r + 900 = 0 \]

8. **Dividing the Equation**:
Dividing the entire equation by 5:
\[ r^2 - 36r + 180 = 0 \]

9. **Using the Quadratic Formula**:
We can use the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -36, c = 180 \):
\[ r = \frac{36 \pm \sqrt{(-36)^2 - 4 \cdot 1 \cdot 180}}{2 \cdot 1} \]
\[ = \frac{36 \pm \sqrt{1296 - 720}}{2} \]
\[ = \frac{36 \pm \sqrt{576}}{2} \]
\[ = \frac{36 \pm 24}{2} \]

10. **Calculating the Roots**:
This gives us two possible values:
\[ r = \frac{60}{2} = 30 \quad \text{(not possible, as it exceeds 10 m)} \]
\[ r = \frac{12}{2} = 6 \]

11. **Conclusion**:
Thus, the distance of point P from the center of shell A is \( r = 6 \, m \).

### Final Answer:
The distance of point P from the center of shell A is **6 meters**.