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 will follow these steps:

### Step 1: Calculate the force \( F_1 \) before the spheres are brought into contact.

Using Coulomb's law, the force between two charges \( q_1 \) and \( q_2 \) separated by a distance \( R \) is given by:

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

Here, \( q_1 = +10 \, \mu C = 10 \times 10^{-6} \, C \) and \( q_2 = -20 \, \mu C = -20 \times 10^{-6} \, C \).

Substituting the values into the formula:

\[ F_1 = k \frac{|(10 \times 10^{-6})(-20 \times 10^{-6})|}{R^2} \]
\[ F_1 = k \frac{200 \times 10^{-12}}{R^2} \]

### Step 2: Calculate the charges after the spheres are brought into contact.

When the two spheres are brought into contact, the total charge is shared equally because they have the same radius. The total charge \( Q \) is:

\[ Q = q_1 + q_2 = 10 \, \mu C + (-20 \, \mu C) = -10 \, \mu C \]

Since both spheres are identical, the charge on each sphere after contact will be:

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

### Step 3: Calculate the force \( F_2 \) after the spheres are separated.

Now, using the new charges \( q' = -5 \, \mu C \) for both spheres, we can calculate \( F_2 \):

\[ F_2 = k \frac{|q' q'|}{R^2} = k \frac{|-5 \times 10^{-6} \times -5 \times 10^{-6}|}{R^2} \]
\[ F_2 = k \frac{25 \times 10^{-12}}{R^2} \]

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

Now we can find the ratio of the forces:

\[ \frac{F_1}{F_2} = \frac{k \frac{200 \times 10^{-12}}{R^2}}{k \frac{25 \times 10^{-12}}{R^2}} \]

The \( k \) and \( R^2 \) terms cancel out:

\[ \frac{F_1}{F_2} = \frac{200 \times 10^{-12}}{25 \times 10^{-12}} = \frac{200}{25} = 8 \]

### Final Answer

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

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

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