Two charges 8 C and -6 C are placed with a distance of separation 'd' between them and exert a force of magnitude F on each other. If a charge 8 C is added to each of these and they are brought nearer by a distance `(d)/(3)`, the magnitude of force between them will be -

Options

A. `(1)/(3) F`
B. `(9)/(4) F`
C. `(3)/(2) F`
D. `(2)/(3) F`

Explanation

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 the formula:

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

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: Calculate the initial force \( F \)

Given:
- \( Q_1 = 8 \, C \)
- \( Q_2 = -6 \, C \)
- Distance \( r = d \)

Using Coulomb's law:

\[ F = k \frac{|8 \times (-6)|}{d^2} = k \frac{48}{d^2} \]

### Step 2: Modify the charges and distance

Now, we add \( 8 \, C \) to each charge:
- New \( Q_1 = 8 + 8 = 16 \, C \)
- New \( Q_2 = -6 + 8 = 2 \, C \)

The new distance between the charges is reduced by \( \frac{d}{3} \):
- New distance \( r = d - \frac{d}{3} = \frac{2d}{3} \)

### Step 3: Calculate the new force \( F' \)

Using the new values in Coulomb's law:

\[ F' = k \frac{|16 \times 2|}{\left(\frac{2d}{3}\right)^2} \]

Calculating the denominator:

\[ \left(\frac{2d}{3}\right)^2 = \frac{4d^2}{9} \]

Now substituting back into the force equation:

\[ F' = k \frac{32}{\frac{4d^2}{9}} = k \frac{32 \times 9}{4d^2} = k \frac{288}{4d^2} = k \frac{72}{d^2} \]

### Step 4: Relate the new force \( F' \) to the original force \( F \)

From the initial force \( F = k \frac{48}{d^2} \), we can express \( k \) in terms of \( F \):

\[ k = \frac{48 F}{d^2} \]

Substituting \( k \) into the new force equation:

\[ F' = \frac{72}{d^2} \cdot \frac{48 F}{d^2} = \frac{72 \cdot 48 F}{d^2} \]

Now we can simplify:

\[ F' = \frac{72}{48} F = \frac{3}{2} F \]

### Conclusion

Thus, the magnitude of the new force between the charges is:

\[ F' = \frac{3}{2} F \]