Two identical coins be 4.5 m apart on a table. They carry similar charges. If the force of repulsion of 40/9 N, then charge on each one is

Options

A. `100muC`
B. `440muC`
C. `110mu C`
D. `550muC`

Explanation

To find the charge on each coin, we can use Coulomb's law, which states that the force between two point charges is given by:

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

Where:
- \( F \) is the force between the charges,
- \( k \) is Coulomb's constant (\( k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.

### Step 1: Identify the given values
- The distance \( r = 4.5 \, \text{m} \)
- The force of repulsion \( F = \frac{40}{9} \, \text{N} \)
- Since both coins are identical and carry similar charges, we can denote the charge on each coin as \( q \).

### Step 2: Substitute the values into Coulomb's law
Since both charges are the same, we can rewrite the equation as:

\[ F = k \frac{q^2}{r^2} \]

Substituting the known values:

\[ \frac{40}{9} = 9 \times 10^9 \frac{q^2}{(4.5)^2} \]

### Step 3: Calculate \( (4.5)^2 \)
Calculating the square of the distance:

\[ (4.5)^2 = 20.25 \]

### Step 4: Substitute \( (4.5)^2 \) back into the equation
Now, we can rewrite the equation:

\[ \frac{40}{9} = 9 \times 10^9 \frac{q^2}{20.25} \]

### Step 5: Rearranging the equation to solve for \( q^2 \)
Multiplying both sides by \( 20.25 \):

\[ 20.25 \cdot \frac{40}{9} = 9 \times 10^9 q^2 \]

Calculating \( 20.25 \cdot \frac{40}{9} \):

\[ 20.25 \cdot \frac{40}{9} = \frac{810}{9} = 90 \]

So we have:

\[ 90 = 9 \times 10^9 q^2 \]

### Step 6: Solve for \( q^2 \)
Dividing both sides by \( 9 \times 10^9 \):

\[ q^2 = \frac{90}{9 \times 10^9} = \frac{10}{10^9} = 10^{-8} \]

### Step 7: Calculate \( q \)
Taking the square root of both sides:

\[ q = \sqrt{10^{-8}} = 10^{-4} \, \text{C} \]

### Step 8: Convert to microcoulombs
To convert coulombs to microcoulombs:

\[ q = 10^{-4} \, \text{C} = 10^{-4} \times 10^6 \, \mu\text{C} = 100 \, \mu\text{C} \]

### Final Answer
The charge on each coin is \( 100 \, \mu\text{C} \).

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