Two point charges +10 muC and +20 muC repel each other with a force of 100N. If a charge of-2 muC is added to each charge, then force between them will become

Options

A. 72N
B. 7.2N
C. 720N
D. 100N

Answer

To solve the problem, we will use Coulomb's law, which states that the force \( F \) between two point charges is given by the formula:

\[ F = k \frac{|Q_1 Q_2|}{d^2} \]

where:
- \( F \) is the force between the charges,
- \( k \) is Coulomb's constant (\( k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges,
- \( d \) is the distance between the charges.

### Step 1: Identify the initial charges and force
The initial charges are:
- \( Q_1 = +10 \, \mu C = 10 \times 10^{-6} \, C \)
- \( Q_2 = +20 \, \mu C = 20 \times 10^{-6} \, C \)

The initial force between them is given as:
- \( F = 100 \, N \)

### Step 2: Write the equation for the initial force
Using Coulomb's law, we can express the initial force as:

\[ 100 = k \frac{|10 \times 10^{-6} \cdot 20 \times 10^{-6}|}{d^2} \]

### Step 3: Calculate the new charges after adding -2 µC
Now, we add -2 µC to each charge:
- New \( Q_1' = 10 \, \mu C - 2 \, \mu C = 8 \, \mu C = 8 \times 10^{-6} \, C \)
- New \( Q_2' = 20 \, \mu C - 2 \, \mu C = 18 \, \mu C = 18 \times 10^{-6} \, C \)

### Step 4: Write the equation for the new force
The new force \( F' \) can be expressed as:

\[ F' = k \frac{|Q_1' Q_2'|}{d^2} = k \frac{|8 \times 10^{-6} \cdot 18 \times 10^{-6}|}{d^2} \]

### Step 5: Relate the new force to the initial force
We can relate the new force to the initial force using the ratio of the charges:

\[ F' = F \cdot \frac{Q_1' Q_2'}{Q_1 Q_2} \]

Substituting the values:

\[ F' = 100 \cdot \frac{(8 \times 10^{-6})(18 \times 10^{-6})}{(10 \times 10^{-6})(20 \times 10^{-6})} \]

### Step 6: Simplify the expression
Calculating the ratio:

\[ F' = 100 \cdot \frac{(8 \cdot 18)}{(10 \cdot 20)} = 100 \cdot \frac{144}{200} = 100 \cdot 0.72 = 72 \, N \]

### Final Answer
Thus, the new force between the charges after adding -2 µC to each is:

\[ F' = 72 \, N \]