Two electrons separated by distance .r. experience a force F between them. The force between a proton and a singly ionized helium atom separated by distance 2r is

Options

A. 4F
B. 2F
C. F/2
D. F/4

Answer

To solve the problem, we need to analyze the forces acting between the charged particles involved. We have two scenarios:

1. The force between two electrons separated by a distance \( r \).
2. The force between a proton and a singly ionized helium atom (which has one electron removed) separated by a distance \( 2r \).

### Step-by-step Solution:

**Step 1: Calculate the force between the two electrons.**

The electrostatic force \( F \) between two charges is given by Coulomb's law:

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

For two electrons, both charges \( q_1 \) and \( q_2 \) are equal to the charge of an electron \( e \). Thus, we can write:

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

**Step 2: Identify the charge of the singly ionized helium atom.**

A singly ionized helium atom (He\(^+\)) consists of two protons and one electron. Therefore, the charge of the proton is \( +e \), and the charge of the singly ionized helium atom is \( +2e \) (from the two protons) and \( -e \) (from the one remaining electron). The total charge of the helium ion is:

\[ q_{He^+} = +e \]

**Step 3: Calculate the force between the proton and the singly ionized helium atom.**

Using Coulomb's law again, the force \( F' \) between the proton and the singly ionized helium atom separated by a distance \( 2r \) is:

\[ F' = k \frac{q_{proton} \cdot q_{He^+}}{(2r)^2} \]

Substituting the charges:

\[ F' = k \frac{e \cdot e}{(2r)^2} = k \frac{e^2}{4r^2} \]

**Step 4: Relate the two forces.**

From Step 1, we have:

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

From Step 3, we have:

\[ F' = k \frac{e^2}{4r^2} \]

Now, we can express \( F' \) in terms of \( F \):

\[ F' = \frac{F}{4} \]

### Conclusion:

The force between the proton and the singly ionized helium atom separated by a distance \( 2r \) is \( \frac{F}{4} \).

### Final Answer:

The force between the proton and the singly ionized helium atom is \( \frac{F}{4} \).

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