The electrostatics force of repulsion between two positively charged ions carrying equal charge is `3.7xx10^(-9)N` when these are separated by a distance of `5Å`. How many electrons are missing from each ion?

Options

A. 20
B. 2
C. 40
D. 4

Explanation

To solve the problem step by step, we will use Coulomb's Law and the known charge of an electron.

### Step 1: Understand the given data
We have two positively charged ions with equal charge, separated by a distance of \(5 \, \text{Å}\) (which is \(5 \times 10^{-10} \, \text{m}\)), and the electrostatic force of repulsion between them is \(3.7 \times 10^{-9} \, \text{N}\).

### Step 2: Use Coulomb's Law
Coulomb's Law states that the force \(F\) between two charges \(q_1\) and \(q_2\) separated by a distance \(r\) is given by:

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

Where \(k\) is Coulomb's constant, approximately \(9 \times 10^9 \, \text{N m}^2/\text{C}^2\).

### Step 3: Substitute the known values
Since both ions have equal charge, we can denote their charge as \(q\). Therefore, the equation becomes:

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

Substituting the values we have:

\[ 3.7 \times 10^{-9} = 9 \times 10^9 \frac{q^2}{(5 \times 10^{-10})^2} \]

### Step 4: Solve for \(q^2\)
First, calculate \( (5 \times 10^{-10})^2 \):

\[ (5 \times 10^{-10})^2 = 25 \times 10^{-20} = 2.5 \times 10^{-19} \]

Now substitute this back into the equation:

\[ 3.7 \times 10^{-9} = 9 \times 10^9 \frac{q^2}{2.5 \times 10^{-19}} \]

Rearranging gives:

\[ q^2 = \frac{3.7 \times 10^{-9} \times 2.5 \times 10^{-19}}{9 \times 10^9} \]

Calculating the right side:

\[ q^2 = \frac{9.25 \times 10^{-28}}{9 \times 10^9} = 1.02778 \times 10^{-37} \]

### Step 5: Find \(q\)
Taking the square root of both sides:

\[ q = \sqrt{1.02778 \times 10^{-37}} \approx 3.2 \times 10^{-19} \, \text{C} \]

### Step 6: Calculate the number of missing electrons
The charge of a single electron \(e\) is approximately \(1.6 \times 10^{-19} \, \text{C}\). The number of electrons \(n\) missing from each ion can be calculated using:

\[ n = \frac{q}{e} \]

Substituting the values we have:

\[ n = \frac{3.2 \times 10^{-19}}{1.6 \times 10^{-19}} = 2 \]

### Final Answer
Thus, the number of electrons missing from each ion is **2**.

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