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 step by step, we will use Coulomb's Law, which states that the force \( F \) between two point charges is given by:

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

where:
- \( F \) is the force between the charges,
- \( k \) is Coulomb's constant,
- \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.

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

The initial force \( F_1 \) between them is given as:
\[ F_1 = 100 \, N \]

### Step 2: Write the equation for the initial force
Using Coulomb's Law, we can express the initial force as:
\[ F_1 = k \frac{Q_1 Q_2}{r^2} \]
Substituting the values:
\[ 100 = k \frac{(10 \times 10^{-6})(20 \times 10^{-6})}{r^2} \]

### Step 3: Modify the charges
Now, we add a charge of \( -2 \, \mu C \) 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 \( F_2 \) between the modified charges is given by:
\[ F_2 = k \frac{Q_1' Q_2'}{r^2} \]
Substituting the new charges:
\[ F_2 = k \frac{(8 \times 10^{-6})(18 \times 10^{-6})}{r^2} \]

### Step 5: Relate the new force to the initial force
We can find the ratio of the new force to the initial force:
\[ \frac{F_2}{F_1} = \frac{(8 \times 10^{-6})(18 \times 10^{-6})}{(10 \times 10^{-6})(20 \times 10^{-6})} \]

### Step 6: Simplify the ratio
Calculating the right side:
\[ \frac{F_2}{100} = \frac{(8 \times 18)}{(10 \times 20)} \]
\[ = \frac{144}{200} = \frac{72}{100} \]

### Step 7: Solve for \( F_2 \)
Now, we can solve for \( F_2 \):
\[ F_2 = 100 \times \frac{72}{100} = 72 \, N \]

### Final Answer
The new force between the charges after adding \( -2 \, \mu C \) to each charge is:
\[ F_2 = 72 \, N \]