What is the Coulomb's force between two `alpha`-particles separated by a distance of `3.2xx10^(-15) m`.

Options

A. `90 N`
B. `45 N`
C. `60 N`
D. `75 N`

Explanation

To solve the problem of finding the Coulomb's force between two alpha particles separated by a distance of \(3.2 \times 10^{-15} \, \text{m}\), we will follow these steps:

### Step 1: Identify the charge of an alpha particle
An alpha particle is essentially a helium nucleus, which consists of 2 protons and 2 neutrons. Since neutrons do not carry a charge, the charge of an alpha particle is due to the 2 protons. The charge of a proton is approximately \(1.6 \times 10^{-19} \, \text{C}\). Therefore, the total charge \(Q\) of an alpha particle is:
\[ Q = 2 \times (1.6 \times 10^{-19} \, \text{C}) = 3.2 \times 10^{-19} \, \text{C} \]

### Step 2: Use Coulomb's Law to calculate the force
Coulomb's Law states that the force \(F\) between two point charges is given by the formula:
\[ F = \frac{k \cdot |Q_1 \cdot Q_2|}{r^2} \]
where:
- \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\),
- \(Q_1\) and \(Q_2\) are the charges of the two particles,
- \(r\) is the distance between the charges.

Since we have two identical alpha particles, we can set \(Q_1 = Q_2 = Q\):
\[ F = \frac{k \cdot Q^2}{r^2} \]

### Step 3: Substitute the known values
Now we will substitute the values we have:
- \(Q = 3.2 \times 10^{-19} \, \text{C}\),
- \(r = 3.2 \times 10^{-15} \, \text{m}\),
- \(k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\).

Substituting these into the formula:
\[ F = \frac{(8.99 \times 10^9) \cdot (3.2 \times 10^{-19})^2}{(3.2 \times 10^{-15})^2} \]

### Step 4: Calculate \(F\)
First, calculate \(Q^2\):
\[ Q^2 = (3.2 \times 10^{-19})^2 = 10.24 \times 10^{-38} \, \text{C}^2 \]

Now calculate \(r^2\):
\[ r^2 = (3.2 \times 10^{-15})^2 = 10.24 \times 10^{-30} \, \text{m}^2 \]

Now substitute these values back into the equation:
\[ F = \frac{(8.99 \times 10^9) \cdot (10.24 \times 10^{-38})}{10.24 \times 10^{-30}} \]

This simplifies to:
\[ F = 8.99 \times 10^9 \cdot 10^{-8} = 8.99 \times 10^1 \, \text{N} = 89.9 \, \text{N} \]

### Step 5: Final Result
Thus, the Coulomb's force between the two alpha particles is approximately:
\[ F \approx 90 \, \text{N} \]