KinexiaLEARN PHYSICS

Electric Field — Inside a Hydrogen Atom (Worked Example)

Hard
Problem

In the Bohr model, the electron orbits at r = 5.29 × 10⁻¹¹ m from the proton (charge +e = 1.602 × 10⁻¹⁹ C). What is the electric field strength at the electron's position?

Given

  • Q = 1.602 × 10⁻¹⁹ C

  • r = 5.29 × 10⁻¹¹ m

  • k = 8.99 × 10⁹ N·m²/C²

Find

The field magnitude E.

Principle

E = kQ/r² applies from power lines down to atoms — the tiny r here makes the field enormous.

Formula

Solution

  1. Substitute: E=(8.99×109)(1.602×10−19)/(5.29×10−11)2E = (8.99 \times 10^9)(1.602 \times 10^{-19})/(5.29 \times 10^{-11})^2

  2. Numerator: 1.440×10−91.440 \times 10^{-9}; denominator: 2.798×10−212.798 \times 10^{-21}

  3. Divide: E=5.15×1011 N/CE = 5.15 \times 10^{11}\ \mathrm{N/C}

5.15 × 10¹¹ N/C toward the proton

Unit check

N·m²/C² × C ÷ m² = N/C — correct.

What this means physically

5.15 × 10¹¹ N/C — hundreds of billions of newtons per coulomb. Atomic fields are colossal, which is why electrons stay bound despite their tiny mass.

Common mistake

Squaring r incorrectly with scientific notation: (5.29 × 10⁻¹¹)² = 27.98 × 10⁻²² = 2.798 × 10⁻²¹ — square the mantissa AND double the exponent.