The force between two small charged spheres having charges of `1xx10^(-7)C and 2xx10^(-7)C` placed 20 cm apart in air is

Options

A. `4.5xx10^(-2)N`
B. `4.5xx10^(-3)N`
C. `5.4xx10^(-2)N`
D. `5.4xx10^(-3)N`

Explanation

To solve the problem of finding the force between two small charged spheres with given charges and distance, we will use Coulomb's Law. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Charge of the first sphere, \( Q_1 = 1 \times 10^{-7} \, \text{C} \)
- Charge of the second sphere, \( Q_2 = 2 \times 10^{-7} \, \text{C} \)
- Distance between the charges, \( r = 20 \, \text{cm} = 0.20 \, \text{m} \)

### Step 2: Write down Coulomb's Law
Coulomb's Law states that the force \( F \) between two point charges is given by:
\[ F = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q_1 Q_2}{r^2} \]
where \( \epsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

### Step 3: Substitute \( \frac{1}{4 \pi \epsilon_0} \)
The value of \( \frac{1}{4 \pi \epsilon_0} \) is approximately \( 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \). Therefore, we can rewrite the equation as:
\[ F = 9 \times 10^9 \cdot \frac{Q_1 Q_2}{r^2} \]

### Step 4: Substitute the values of \( Q_1 \), \( Q_2 \), and \( r \)
Now, substituting the values we have:
\[ F = 9 \times 10^9 \cdot \frac{(1 \times 10^{-7})(2 \times 10^{-7})}{(0.20)^2} \]

### Step 5: Calculate \( r^2 \)
Calculate \( r^2 \):
\[ r^2 = (0.20)^2 = 0.04 \, \text{m}^2 \]

### Step 6: Calculate \( Q_1 Q_2 \)
Calculate \( Q_1 Q_2 \):
\[ Q_1 Q_2 = (1 \times 10^{-7})(2 \times 10^{-7}) = 2 \times 10^{-14} \, \text{C}^2 \]

### Step 7: Substitute back into the force equation
Now substitute \( Q_1 Q_2 \) and \( r^2 \) back into the force equation:
\[ F = 9 \times 10^9 \cdot \frac{2 \times 10^{-14}}{0.04} \]

### Step 8: Simplify the expression
Calculate the fraction:
\[ \frac{2 \times 10^{-14}}{0.04} = 5 \times 10^{-13} \]
Now substitute this back:
\[ F = 9 \times 10^9 \cdot 5 \times 10^{-13} \]

### Step 9: Calculate the final force
Now calculate the force:
\[ F = 45 \times 10^{-4} \, \text{N} = 4.5 \times 10^{-3} \, \text{N} \]

### Final Answer
The force between the two charged spheres is:
\[ F = 4.5 \times 10^{-3} \, \text{N} \]

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