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 equlibrium state:

Options

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

Explanation

To solve the problem of finding the position of a third charge \( q \) between two point charges \( +9e \) and \( +e \) that are 16 cm apart, we can follow these steps:

### Step 1: Define the Positions
Let the position of charge \( +9e \) be at point A and the position of charge \( +e \) be at point B. The distance between them is given as 16 cm. We will place the charge \( q \) at a distance \( r \) from charge \( +9e \) and therefore at a distance \( 16 - r \) from charge \( +e \).

### Step 2: Set Up the Force Equations
For the system to be in equilibrium, the net force acting on charge \( q \) must be zero. This means that the force exerted on \( q \) by charge \( +9e \) must be equal to the force exerted on \( q \) by charge \( +e \).

Using Coulomb's law, the forces can be expressed as:
- Force on \( q \) due to \( +9e \):
\[ F_1 = k \frac{(9e)q}{r^2} \]
- Force on \( q \) due to \( +e \):
\[ F_2 = k \frac{eq}{(16 - r)^2} \]

### Step 3: Set the Forces Equal
For equilibrium:
\[ F_1 = F_2 \]
This gives us:
\[ k \frac{(9e)q}{r^2} = k \frac{eq}{(16 - r)^2} \]
We can cancel \( k \) and \( q \) from both sides (assuming \( q \neq 0 \)):
\[ \frac{9e}{r^2} = \frac{e}{(16 - r)^2} \]

### Step 4: Simplify the Equation
We can simplify the equation further by canceling \( e \):
\[ \frac{9}{r^2} = \frac{1}{(16 - r)^2} \]
Cross-multiplying gives:
\[ 9(16 - r)^2 = r^2 \]

### Step 5: Expand and Rearrange
Expanding the left-hand side:
\[ 9(256 - 32r + r^2) = r^2 \]
This simplifies to:
\[ 2304 - 288r + 9r^2 = r^2 \]
Rearranging gives:
\[ 8r^2 - 288r + 2304 = 0 \]

### Step 6: Solve the Quadratic Equation
Now we can use the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 8 \), \( b = -288 \), and \( c = 2304 \):
\[ r = \frac{288 \pm \sqrt{(-288)^2 - 4 \cdot 8 \cdot 2304}}{2 \cdot 8} \]
Calculating the discriminant:
\[ (-288)^2 = 82944 \]
\[ 4 \cdot 8 \cdot 2304 = 73728 \]
Thus:
\[ r = \frac{288 \pm \sqrt{82944 - 73728}}{16} \]
\[ r = \frac{288 \pm \sqrt{9216}}{16} \]
\[ \sqrt{9216} = 96 \]
So:
\[ r = \frac{288 \pm 96}{16} \]
Calculating the two possible values for \( r \):
1. \( r = \frac{384}{16} = 24 \) cm
2. \( r = \frac{192}{16} = 12 \) cm

### Step 7: Determine the Valid Position
Since \( r \) must be between the two charges (which are 0 cm and 16 cm), the valid position for charge \( q \) is:
\[ r = 12 \text{ cm} \]

Thus, the charge \( q \) should be placed **12 cm from charge \( +9e \)**.