Two identical conducting sphere carrying different charges attact each other with a force F when placed in air medium at a distance `d` apart. The spheres are brought into contact and then taken to their original positions. Now, the two sphere repel each other with a force whole magnitude is equal to the initial attractive force. The ratio between initial charges on the spheres is

Options

A. `- (3 + sqrt8)` only
B. `-3 + sqrt8` only
C. `-(3 + sqrt8)` or `(-3 + sqrt8)`
D. `+ sqrt3`

Answer

To solve the problem step by step, we will analyze the situation involving two identical conducting spheres with different charges.

### Step 1: Understand the Initial Condition
Let the charges on the two spheres be \( Q_1 \) and \( Q_2 \). The force of attraction between them when they are at a distance \( d \) apart is given by Coulomb's Law:
\[ F = k \frac{|Q_1 Q_2|}{d^2} \]
where \( k \) is Coulomb's constant.

### Step 2: Bring the Spheres into Contact
When the two spheres are brought into contact, they will share their total charge equally because they are identical conductors. The total charge \( Q \) is given by:
\[ Q = Q_1 + Q_2 \]
After contact, each sphere will have a charge:
\[ Q' = \frac{Q_1 + Q_2}{2} \]

### Step 3: Calculate the New Force After Contact
After being separated again, the charges on the spheres will be \( Q' \) and \( Q' \). The force of repulsion between them at the same distance \( d \) is:
\[ F' = k \frac{(Q')^2}{d^2} \]
Substituting for \( Q' \):
\[ F' = k \frac{\left(\frac{Q_1 + Q_2}{2}\right)^2}{d^2} = k \frac{(Q_1 + Q_2)^2}{4d^2} \]

### Step 4: Set the Forces Equal
According to the problem, the magnitude of the repulsive force \( F' \) is equal to the initial attractive force \( F \):
\[ k \frac{|Q_1 Q_2|}{d^2} = k \frac{(Q_1 + Q_2)^2}{4d^2} \]
Cancelling \( k \) and \( d^2 \) from both sides gives:
\[ |Q_1 Q_2| = \frac{(Q_1 + Q_2)^2}{4} \]

### Step 5: Rearranging the Equation
Rearranging the equation, we have:
\[ 4|Q_1 Q_2| = (Q_1 + Q_2)^2 \]
Expanding the right side:
\[ 4|Q_1 Q_2| = Q_1^2 + 2Q_1Q_2 + Q_2^2 \]
This can be rearranged to:
\[ Q_1^2 + Q_2^2 - 2Q_1Q_2 - 4|Q_1 Q_2| = 0 \]

### Step 6: Substitute \( Q_2 = -k Q_1 \)
Assuming \( Q_2 = -k Q_1 \) (since they attract each other), we can substitute this into the equation:
\[ Q_1^2 + (-k Q_1)^2 - 2Q_1(-k Q_1) - 4|Q_1(-k Q_1)| = 0 \]
This simplifies to:
\[ Q_1^2 + k^2 Q_1^2 + 2k Q_1^2 - 4k Q_1^2 = 0 \]
Combining like terms gives:
\[ (1 + k^2 - 2k)Q_1^2 = 0 \]
This implies:
\[ 1 + k^2 - 2k = 0 \]
Solving this quadratic equation for \( k \):
\[ k^2 - 2k + 1 = 0 \implies (k - 1)^2 = 0 \implies k = 1 \]

### Step 7: Find the Ratio of Charges
The ratio of the initial charges \( \frac{Q_1}{Q_2} \) can be derived from \( k = \frac{Q_1}{-Q_2} \):
\[ \frac{Q_1}{Q_2} = 1 \implies Q_1 = -Q_2 \]
Thus, the ratio of the charges is:
\[ \frac{Q_1}{Q_2} = 1 \]

### Final Answer
The ratio between the initial charges on the spheres is \( 1:1 \).

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