Two small conducting spheres of equal radius have charges `+ 10 muC` and `-20 muC` respectively and placed at a distance R from each other. They experience force `F_(1)`. If they are brought in contact and separated to the same distance, they experience force `F_(2)`. The ratio of `F_(1)` to `F_(2)` is

Options

A. `1 : 8`
B. `-8 : 1`
C. `1 : 2`
D. `-2 : 1`

Answer

To solve the problem, we need to calculate the forces \( F_1 \) and \( F_2 \) experienced by the two spheres before and after they are brought into contact.

### Step 1: Calculate \( F_1 \)

Given:
- Charge of sphere 1, \( q_1 = +10 \, \mu C = 10 \times 10^{-6} \, C \)
- Charge of sphere 2, \( q_2 = -20 \, \mu C = -20 \times 10^{-6} \, C \)
- Distance between the spheres, \( R \)

The electrostatic force \( F_1 \) between the two charges can be calculated using Coulomb's Law:

\[ F_1 = k \frac{|q_1 \cdot q_2|}{R^2} \]

Where \( k \) is Coulomb's constant, \( k = 9 \times 10^9 \, N \cdot m^2/C^2 \).

Substituting the values:

\[ F_1 = 9 \times 10^9 \frac{|10 \times 10^{-6} \cdot (-20) \times 10^{-6}|}{R^2} \]

Calculating:

\[ F_1 = 9 \times 10^9 \frac{200 \times 10^{-12}}{R^2} = \frac{1800 \times 10^{-3}}{R^2} = \frac{1.8}{R^2} \, N \]

### Step 2: Calculate the total charge after contact

When the two spheres are brought into contact, they share their charges. The total charge \( Q \) is:

\[ Q = q_1 + q_2 = 10 \times 10^{-6} + (-20 \times 10^{-6}) = -10 \times 10^{-6} \, C \]

Since they are identical spheres, the charge on each sphere after separation will be:

\[ q' = \frac{Q}{2} = \frac{-10 \times 10^{-6}}{2} = -5 \times 10^{-6} \, C \]

### Step 3: Calculate \( F_2 \)

Now, we calculate the force \( F_2 \) when the spheres are separated again at the same distance \( R \):

\[ F_2 = k \frac{|q' \cdot q'|}{R^2} \]

Substituting the values:

\[ F_2 = 9 \times 10^9 \frac{|-5 \times 10^{-6} \cdot -5 \times 10^{-6}|}{R^2} \]

Calculating:

\[ F_2 = 9 \times 10^9 \frac{25 \times 10^{-12}}{R^2} = \frac{225 \times 10^{-3}}{R^2} = \frac{0.225}{R^2} \, N \]

### Step 4: Find the ratio \( \frac{F_1}{F_2} \)

Now, we can find the ratio \( \frac{F_1}{F_2} \):

\[ \frac{F_1}{F_2} = \frac{\frac{1.8}{R^2}}{\frac{0.225}{R^2}} = \frac{1.8}{0.225} \]

Calculating the ratio:

\[ \frac{F_1}{F_2} = \frac{1.8 \div 0.225} = 8 \]

Thus, the ratio of \( F_1 \) to \( F_2 \) is:

\[ \frac{F_1}{F_2} = 8:1 \]

### Final Answer:
The ratio of \( F_1 \) to \( F_2 \) is \( 8:1 \).

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