What is the electric field?

How it works

With k = 1 ÷ (4πε₀) and ε₀ = 8.8541878188 × 10⁻¹² F/m (CODATA 2022), for a point charge q in a vacuum:

• Field: E = k × |q| ÷ r² at distance r. It points away from a positive charge and toward a negative one.
• Distance for a field strength: r = √(k × |q| ÷ E).
• Charge for a field at a distance: |q| = E × r² ÷ k, shown in coulombs and in proton charges (|q| ÷ e). The field alone does not give the sign: q is positive if the field points away from it and negative if it points toward it.
• Potential: V = k × q ÷ r, in volts, with V = 0 far away (negative for a negative charge). It is shown when the source charge is typed.
• Coulomb force on a second charge Q at that point: F = |Q| × E = k × |q × Q| ÷ r², in newtons. It is repulsive when q and Q have the same sign and attractive when the signs differ.

The text outputs read exactly: the field direction is Away from the charge or Toward the charge; the force is Repulsive: the charges push apart, Attractive: the charges pull together, or None: the second charge is 0 (with a force of 0). With a charge found from a field, the page gives the force size but no direction or force text.

Units (exact): 1 mC = 10⁻³ C; 1 µC = 10⁻⁶ C; 1 nC = 10⁻⁹ C; 1 pC = 10⁻¹² C; 1 e = 1.602176634 × 10⁻¹⁹ C; 1 nm = 10⁻⁹ m; 1 µm = 10⁻⁶ m; 1 mm = 0.001 m; 1 cm = 0.01 m; 1 km = 1,000 m; 1 N/C = 1 V/m; 1 kN/C = 1,000 N/C; 1 MN/C = 10⁶ N/C.

Rules

• Charges are numbers from −10¹² to 10¹² in the unit chosen beside them; distance from 10⁻¹⁵ m to 10¹⁵ m (limits in metres, a value in another unit converted first); field strength from 10⁻¹² to 10²⁴ in its unit. Limits include their ends.
• A source charge of 0 makes no field: there is no answer. The second charge is optional; a second charge of 0 feels no force.
• A result too large or too small to show as a number gives no answer (a tiny charge far away can make a field below the smallest double).

Output format

The answer line reads E = a N/C, r = a m or |q| = a C for the value found. Its number has 6 significant figures, rounded half up, with trailing zeros dropped; from 10⁶ up and below 10⁻⁶ it is written as a × 10ⁿ (E = 4.101 × 10¹² N/C), and otherwise with thousands commas. Every other number shows to 6 significant figures in the same way, but as a × 10ⁿ only below 10⁻⁶ or from 10¹⁵ up; negative numbers show with a minus sign (−). Typed values are read back as the exact decimals you typed, and every unit size and ε₀ are exact, so |q| ÷ (4ε₀r²) is an exact fraction, divided by π once. A distance is the square root of the exact |q| ÷ (4ε₀E), rounded once, divided by √π.

Worked examples by hand

A helium nucleus. OpenStax Example 5.3: q = 2e = 3.204353268 × 10⁻¹⁹ C at r = 26.5 × 10⁻¹² m: E = 3.204353268 × 10⁻¹⁹ ÷ (4π × 8.8541878188 × 10⁻¹² × (2.65 × 10⁻¹¹)²) = 4.10100 × 10¹² N/C, shown as E = 4.101 × 10¹² N/C, pointing away from the nucleus (OpenStax: 4.1 × 10¹² N/C).

Hydrogen. OpenStax Example 5.1: +e and −e at 5.29 × 10⁻¹¹ m: F = (1.602176634 × 10⁻¹⁹)² ÷ (4π × 8.8541878188 × 10⁻¹² × (5.29 × 10⁻¹¹)²) = 8.24424 × 10⁻⁸ N, Attractive: the charges pull together (OpenStax: 8.25 × 10⁻⁸ N).

One nanocoulomb. k = 1 ÷ (4π × 8.8541878188 × 10⁻¹²) = 8.98755 × 10⁹, so 1 nC at 1 m gives E = 8.98755 N/C and V = 8.98755 V.

A distance. Where does a 1 µC charge make 10,000 N/C? r = √(8.98755 × 10⁹ × 10⁻⁶ ÷ 10,000) = √0.898755 = 0.948027 m.

A charge. What charge makes 10,000 N/C at 1 m? |q| = 10,000 × 1² ÷ 8.98755 × 10⁹ = 1.11265 × 10⁻⁶ C.