Two identical metals balls with charges `+2Q` and `-Q` are separated by some distance and exert a force `F` on each other . They are joined by a conducting wire , which is then removed. The force between them will now be

Options

A. `F`
B. `(F)/(2)`
C. `(F)/(4)`
D. `(F)/(8)`

Explanation

To solve the problem step by step, we will analyze the situation before and after the two identical metal balls are connected by a conducting wire.

### Step 1: Initial Force Calculation
We start with two identical metal balls with charges \( +2Q \) and \( -Q \). The force \( F \) between them can be calculated using Coulomb's law:

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

Where:
- \( k \) is the electrostatic constant,
- \( Q_1 = +2Q \),
- \( Q_2 = -Q \),
- \( R \) is the distance between the two balls.

Substituting the values, we get:

\[ F = k \frac{|(2Q)(-Q)|}{R^2} = k \frac{2Q^2}{R^2} \]

### Step 2: Connecting the Balls with a Conducting Wire
When the two balls are connected by a conducting wire, charges will redistribute until they reach the same potential. The total charge before connecting the wire is:

\[ Q_{\text{total}} = +2Q + (-Q) = Q \]

Since the balls are identical, the charge will be equally distributed between them:

\[ \text{Charge on each ball after connection} = \frac{Q}{2} \]

### Step 3: Force Calculation After Charge Redistribution
Now, after the wire is removed, each ball has a charge of \( \frac{Q}{2} \). We need to calculate the new force \( F' \) between the two balls:

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

Where:
- \( Q_1' = \frac{Q}{2} \),
- \( Q_2' = \frac{Q}{2} \).

Substituting these values, we get:

\[ F' = k \frac{|\left(\frac{Q}{2}\right)\left(\frac{Q}{2}\right)|}{R^2} = k \frac{Q^2/4}{R^2} = \frac{k Q^2}{4 R^2} \]

### Step 4: Relating the New Force to the Initial Force
We can express \( F' \) in terms of the initial force \( F \):

From the initial force \( F = k \frac{2Q^2}{R^2} \), we can write:

\[ F = k \frac{2Q^2}{R^2} \]

Now, we can express \( F' \) as:

\[ F' = \frac{1}{4} \cdot \frac{1}{2} F = \frac{F}{8} \]

### Conclusion
Thus, the force between the two balls after they are connected by a conducting wire and then separated is:

\[ F' = \frac{F}{8} \]