Question

The nucleus of helium atom contains two protons that are separated by a distance 3.0 $\times$ 10⁻¹⁵ m. The magnitude of the electrostatic force that each proton exerts on the other is

• A. 20.6 N
• B. 25.6 N
• C. 15.6 N
• D. 12.6 N

Hint:

When squaring numbers in scientific notation, remember to square the base number normally ($3^2 = 9$) and multiply the exponent power by 2 ($10^{-15 \times 2} = 10^{-30}$). Keeping your exponent tracks separate makes verification much easier!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The interaction between two stationary electric charges is described by Coulomb's Law. In this problem, we are looking at two protons, which are both positively charged particles. Because they have the same sign of charge, they will exert a repulsive electrostatic force on each other. Despite the extremely small distances found within an atomic nucleus, the fundamental law of electrostatics still applies to the point charges.
Step 2: Key Formula or Approach:
1. Coulomb's Law: $F = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2}$.
2. The value of the electrostatic constant $k = \frac{1}{4\pi\varepsilon_0} \approx 9 \times 10^9 \text{ N m}^2/\text{C}^2$.
3. The charge of a proton $q = e \approx 1.6 \times 10^{-19} \text{ C}$.
4. The separation distance $r = 3.0 \times 10^{-15} \text{ m}$.
Step 3: Detailed Explanation:

We begin by substituting the known values into the Coulomb's Law expression: $F = \frac{(9 \times 10^9) \times (1.6 \times 10^{-19}) \times (1.6 \times 10^{-19})}{(3.0 \times 10^{-15})^2}$.

First, calculate the square of the proton charge: $(1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \text{ C}^2$.

Next, calculate the square of the distance: $(3.0 \times 10^{-15})^2 = 9.0 \times 10^{-30} \text{ m}^2$.

Now, combine these into the force formula: $F = \frac{9 \times 10^9 \times 2.56 \times 10^{-38}}{9 \times 10^{-30}}$.

We can cancel the number 9 from the numerator and denominator, which simplifies our calculation significantly.

We are left with $F = 2.56 \times \frac{10^9 \times 10^{-38}}{10^{-30}}$.

Using the rules of exponents, we add and subtract the powers of 10: $9 + (-38) - (-30) = 9 - 38 + 30 = 1$.

This results in $F = 2.56 \times 10^1 = 25.6 \text{ N}$.

Step 4: Final Answer:
The magnitude of the electrostatic force is 25.6 N.