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: Understand the Initial Charges
Initially, we have two identical metal balls:
- Ball 1 has a charge of \( +2Q \)
- Ball 2 has a charge of \( -Q \)

### Step 2: Calculate the Initial Force
The force \( F \) between two charges is given by Coulomb's law:
\[ F = k \frac{|Q_1 Q_2|}{R^2} \]
where \( k \) is Coulomb's constant, \( Q_1 \) and \( Q_2 \) are the charges, and \( R \) is the distance between them.

Substituting the initial charges:
\[ F = k \frac{|(2Q)(-Q)|}{R^2} = k \frac{2Q^2}{R^2} \]

### Step 3: Connecting the Balls with a Conducting Wire
When the two balls are connected by a conducting wire, charge will redistribute between them until they reach the same potential. The total charge before connecting is:
\[ Q_{\text{total}} = +2Q + (-Q) = +Q \]
Since the balls are identical, the charge will be evenly distributed:
\[ Q_1' = Q_2' = \frac{Q_{\text{total}}}{2} = \frac{Q}{2} \]

### Step 4: Calculate the New Charges
After connecting the wire, both balls will have a charge of:
- Ball 1: \( Q_1' = \frac{Q}{2} \)
- Ball 2: \( Q_2' = \frac{Q}{2} \)

### Step 5: Calculate the New Force
Now, we can calculate the new force \( F' \) between the two balls using the new charges:
\[ F' = k \frac{|Q_1' Q_2'|}{R^2} = k \frac{\left(\frac{Q}{2}\right)\left(\frac{Q}{2}\right)}{R^2} = k \frac{\frac{Q^2}{4}}{R^2} = k \frac{Q^2}{4R^2} \]

### Step 6: Relate the New Force to the Initial Force
We can express the new force in terms of the initial force \( F \):
\[ F' = \frac{1}{4} \cdot \frac{1}{2} \cdot F = \frac{1}{8} F \]

### Conclusion
The new force between the two balls after they are connected by a conducting wire and then separated again is:
\[ F' = \frac{F}{8} \]