Two point charges `+9e` and `+e` are kept `16 cm`. Apart from each other. Where should a third charge `q` be placed between them so that the system is in equilibrium state:

Options

A. `24 cm` form `+9e`
B. `12 cm` from `+9e`
C. `24 cm` from `+e`
D. `12 cm` from `+e`

Explanation

To determine where to place a third charge \( q \) between two point charges \( +9e \) and \( +e \) such that the system is in equilibrium, we can follow these steps:

### Step 1: Understand the Configuration
We have two charges:
- Charge \( Q_1 = +9e \) located at point A.
- Charge \( Q_2 = +e \) located at point B.
- The distance between A and B is \( 16 \, \text{cm} \).

We need to find the position \( r \) from charge \( Q_1 \) where the third charge \( q \) should be placed.

### Step 2: Set Up the Problem
Let the distance from \( Q_1 \) to charge \( q \) be \( r \). Consequently, the distance from \( Q_2 \) to charge \( q \) will be \( 16 - r \).

### Step 3: Apply Coulomb's Law
For the charge \( q \) to be in equilibrium, the net force acting on it due to both charges must be zero. The forces acting on \( q \) due to \( Q_1 \) and \( Q_2 \) can be expressed as follows:

- The force \( F_1 \) on charge \( q \) due to \( Q_1 \) is given by:
\[ F_1 = \frac{k \cdot |q \cdot 9e|}{r^2} \]

- The force \( F_2 \) on charge \( q \) due to \( Q_2 \) is given by:
\[ F_2 = \frac{k \cdot |q \cdot e|}{(16 - r)^2} \]

### Step 4: Set Forces Equal
For equilibrium, we set \( F_1 \) equal to \( F_2 \):
\[ \frac{k \cdot |q \cdot 9e|}{r^2} = \frac{k \cdot |q \cdot e|}{(16 - r)^2} \]

Since \( k \) and \( |q| \) are common on both sides, they can be canceled out:
\[ \frac{9e}{r^2} = \frac{e}{(16 - r)^2} \]

### Step 5: Simplify the Equation
We can simplify the equation:
\[ 9(16 - r)^2 = r^2 \]

### Step 6: Expand and Rearrange
Expanding the left side:
\[ 9(256 - 32r + r^2) = r^2 \]
\[ 2304 - 288r + 9r^2 = r^2 \]
Rearranging gives:
\[ 8r^2 - 288r + 2304 = 0 \]

### Step 7: Solve the Quadratic Equation
We can use the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 8 \), \( b = -288 \), and \( c = 2304 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-288)^2 - 4 \cdot 8 \cdot 2304 \]
\[ = 82944 - 73728 = 9216 \]

Now, applying the quadratic formula:
\[ r = \frac{288 \pm \sqrt{9216}}{16} \]
Calculating \( \sqrt{9216} = 96 \):
\[ r = \frac{288 \pm 96}{16} \]

This gives us two possible values for \( r \):
1. \( r = \frac{384}{16} = 24 \, \text{cm} \)
2. \( r = \frac{192}{16} = 12 \, \text{cm} \)

### Step 8: Determine Valid Position
Since \( r \) must be between \( 0 \) and \( 16 \) cm, the valid solution is:
\[ r = 12 \, \text{cm} \]

Thus, the third charge \( q \) should be placed \( 12 \, \text{cm} \) from the charge \( +9e \).

### Final Answer
The charge \( q \) should be placed \( 12 \, \text{cm} \) from the charge \( +9e \).

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