Two charges of magnitude `5 nC and -2 nC` are placed at points (2cm,0,0) and (x cm,0,0) in a region of space. Where there is no other external field. If the electrostatic potential energy of the system is `-0.5 muJ`. What is the value of x ?

Options

A. 20 cm
B. 80 cm
C. 4 cm
D. 16 cm

Explanation

To find the value of \( x \) for the given electrostatic potential energy of the system, we can follow these steps:

### Step 1: Understand the Problem
We have two charges:
- \( Q_1 = 5 \, \text{nC} = 5 \times 10^{-9} \, \text{C} \) located at \( (2 \, \text{cm}, 0, 0) \)
- \( Q_2 = -2 \, \text{nC} = -2 \times 10^{-9} \, \text{C} \) located at \( (x \, \text{cm}, 0, 0) \)

The electrostatic potential energy \( U \) of the system is given as \( -0.5 \, \mu\text{J} = -0.5 \times 10^{-6} \, \text{J} \).

### Step 2: Write the Formula for Electrostatic Potential Energy
The formula for the electrostatic potential energy \( U \) between two point charges is given by:
\[ U = k \frac{Q_1 Q_2}{R} \]
where \( k \) is Coulomb's constant (\( k \approx 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)) and \( R \) is the distance between the charges.

### Step 3: Calculate the Distance \( R \)
The distance \( R \) between the two charges is:
\[ R = |x - 2| \, \text{cm} = |x - 2| \times 10^{-2} \, \text{m} \]

### Step 4: Substitute Values into the Formula
Substituting the values into the potential energy formula:
\[ -0.5 \times 10^{-6} = 9 \times 10^9 \frac{(5 \times 10^{-9})(-2 \times 10^{-9})}{|x - 2| \times 10^{-2}} \]

### Step 5: Simplify the Equation
Calculating the numerator:
\[ (5 \times 10^{-9})(-2 \times 10^{-9}) = -10 \times 10^{-18} = -1 \times 10^{-17} \]
Thus, the equation becomes:
\[ -0.5 \times 10^{-6} = 9 \times 10^9 \frac{-1 \times 10^{-17}}{|x - 2| \times 10^{-2}} \]
This simplifies to:
\[ -0.5 \times 10^{-6} = \frac{-9 \times 10^{-8}}{|x - 2|} \]

### Step 6: Remove the Negative Signs
Removing the negative signs from both sides, we have:
\[ 0.5 \times 10^{-6} = \frac{9 \times 10^{-8}}{|x - 2|} \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ 0.5 \times 10^{-6} |x - 2| = 9 \times 10^{-8} \]

### Step 8: Solve for \( |x - 2| \)
Dividing both sides by \( 0.5 \times 10^{-6} \):
\[ |x - 2| = \frac{9 \times 10^{-8}}{0.5 \times 10^{-6}} = \frac{9}{0.5} \times 10^{-2} = 18 \times 10^{-2} \]
Thus,
\[ |x - 2| = 0.18 \, \text{m} = 18 \, \text{cm} \]

### Step 9: Solve for \( x \)
This gives us two cases:
1. \( x - 2 = 18 \) → \( x = 20 \, \text{cm} \)
2. \( x - 2 = -18 \) → \( x = -16 \, \text{cm} \) (not physically meaningful in this context)

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{20 \, \text{cm}} \]