Determine the electric field produced by a helium nucleus at a distance of 1 Å from it.

Answer

To determine the electric field produced by a helium nucleus at a distance of 1 Å (angstrom), we will follow these steps:

### Step 1: Identify the Charge of the Helium Nucleus
The helium nucleus consists of 2 protons and 2 neutrons. Since neutrons do not carry any charge, the total charge \( Q \) of the helium nucleus is due to the protons only.

\[ Q = 2 \times e \]

Where \( e \) is the elementary charge, approximately \( 1.6 \times 10^{-19} \) coulombs.

### Step 2: Calculate the Total Charge
Substituting the value of \( e \):

\[ Q = 2 \times (1.6 \times 10^{-19}) = 3.2 \times 10^{-19} \text{ coulombs} \]

### Step 3: Determine the Distance
The distance \( r \) from the nucleus at which we want to calculate the electric field is given as 1 Å.

\[ r = 1 \text{ Å} = 1 \times 10^{-10} \text{ meters} \]

### Step 4: Use the Formula for Electric Field
The electric field \( E \) due to a point charge is given by the formula:

\[ E = \frac{k \cdot Q}{r^2} \]

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

### Step 5: Substitute the Values into the Formula
Now we will substitute the values of \( k \), \( Q \), and \( r \) into the formula:

\[ E = \frac{9 \times 10^9 \cdot (3.2 \times 10^{-19})}{(1 \times 10^{-10})^2} \]

### Step 6: Calculate \( r^2 \)
Calculating \( r^2 \):

\[ r^2 = (1 \times 10^{-10})^2 = 1 \times 10^{-20} \text{ m}^2 \]

### Step 7: Substitute \( r^2 \) Back into the Formula
Now substituting \( r^2 \) back into the electric field formula:

\[ E = \frac{9 \times 10^9 \cdot (3.2 \times 10^{-19})}{1 \times 10^{-20}} \]

### Step 8: Simplify the Expression
Calculating the numerator:

\[ 9 \times 10^9 \cdot 3.2 \times 10^{-19} = 28.8 \times 10^{-10} \text{ N/C} \]

Now, dividing by \( 1 \times 10^{-20} \):

\[ E = 28.8 \times 10^{10} \text{ N/C} \]

### Step 9: Final Result
Thus, the electric field \( E \) at a distance of 1 Å from the helium nucleus is:

\[ E = 2.88 \times 10^{11} \text{ N/C} \]

### Step 10: Direction of the Electric Field
Since the helium nucleus is positively charged, the direction of the electric field will be radially outward from the nucleus.

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