How Far Electrons Are From the Nucleus, Scaled to Earth
If Earth were scaled up to represent an atomic nucleus, an atom's electron cloud would extend roughly 1.3 billion km out, far beyond the Moon and into the outer solar system. Electrons have no single fixed distance because their positions are described by probability clouds rather than orbits.
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Scaling the atom to Earth
Using Earth as the nucleus, the electrons of an atom lie much farther out than first estimated. An earlier answer suggested 30 to 40 times the Moon's distance, but that figure was far too small.
The atom's size is set by its electron cloud. A typical atom is about 100,000 times wider than its nucleus. For hydrogen, the atom's radius is roughly 0.5 × 10⁻¹⁰ m (about 53 picometers, the Bohr radius), while the nucleus is around 1 femtometer (10⁻¹⁵ m).
Comparing with the Moon and the solar system
Scaling the nucleus up to Earth's diameter (about 12,700 km) makes the atom roughly 100,000 times that size, or about 1.3 billion km. The Moon orbits about 384,000 km from Earth, so the electron cloud reaches about 3,400 times farther than the Moon.
This distance lies in the outer solar system, past Jupiter (about 780 million km from the Sun) and near Saturn's distance.
Why there is no single distance
Electrons do not orbit like planets. Their positions are described by probability clouds, so the figure above is the approximate radius where electrons are typically found, not an exact distance for any one electron.