{
  "id": "electric-field",
  "version": "7d18eb014661",
  "status": "published",
  "name": "Electric Field Calculator",
  "question": "What is the electric field?",
  "summary": "Electric field calculator: E = kq ÷ r² of a point charge at a distance, with its direction, the potential and the Coulomb force on a second charge; or the distance or charge for a field.",
  "category": "physics",
  "subcategory": "electricity",
  "url": "https://www.acalculator.org/physics/electric-field-calculator",
  "markdown": "https://www.acalculator.org/physics/electric-field-calculator.md",
  "kind": "function",
  "method": "E = k|q| ÷ r² with k = 1 ÷ (4πε₀); r = √(k|q| ÷ E); |q| = E r² ÷ k; V = kq ÷ r; F = k|qQ| ÷ r² = |Q| E.",
  "assumptions": [
    "A point charge (or a sphere of charge, seen from outside it) alone in a vacuum; ε₀ = 8.8541878188 × 10⁻¹² F/m.",
    "The second charge does not move the first; the potential is 0 far away."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "Which value to work out from E = kq ÷ r².",
        "type": "string",
        "enum": [
          "field",
          "distance",
          "charge"
        ]
      },
      "q": {
        "title": "Source charge (q)",
        "description": "The point charge that makes the field; negative for a negative charge.",
        "type": "number",
        "minimum": -1000000000000,
        "maximum": 1000000000000
      },
      "qu": {
        "title": "Charge unit",
        "description": "The unit of the charge.",
        "type": "string",
        "enum": [
          "C",
          "mC",
          "uC",
          "nC",
          "pC",
          "e"
        ]
      },
      "r": {
        "title": "Distance (r)",
        "description": "The distance from the charge to the point.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-15,
        "maximum": 1000000000000000
      },
      "E": {
        "title": "Field strength (E)",
        "description": "The size of the electric field at the point, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1e+24
      },
      "Eu": {
        "title": "Field unit",
        "description": "The unit of the field strength.",
        "type": "string",
        "enum": [
          "NC",
          "kNC",
          "MNC"
        ]
      },
      "q2": {
        "title": "Second charge (Q)",
        "description": "Optional: a charge placed at the point, to find the Coulomb force on it.",
        "type": "number",
        "minimum": -1000000000000,
        "maximum": 1000000000000
      },
      "q2u": {
        "title": "Second charge unit",
        "description": "The unit of the charge.",
        "type": "string",
        "enum": [
          "C",
          "mC",
          "uC",
          "nC",
          "pC",
          "e"
        ]
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The value worked out, to 6 significant figures.",
      "format": "text"
    },
    "field": {
      "label": "Electric field (N/C)",
      "description": "E = k|q| ÷ r², in newtons per coulomb (the same as volts per metre).",
      "format": "number"
    },
    "direction": {
      "label": "Field direction",
      "description": "Away from a positive charge, toward a negative one.",
      "format": "text"
    },
    "distance": {
      "label": "Distance (m)",
      "description": "r, in metres.",
      "format": "number"
    },
    "charge": {
      "label": "Charge size (C)",
      "description": "|q| = E r² ÷ k, in coulombs.",
      "format": "number"
    },
    "charges": {
      "label": "Charge size (e)",
      "description": "|q| in proton charges, e = 1.602176634 × 10⁻¹⁹ C.",
      "format": "number"
    },
    "potential": {
      "label": "Electric potential (V)",
      "description": "V = kq ÷ r, in volts, taking 0 far away.",
      "format": "number"
    },
    "force": {
      "label": "Coulomb force (N)",
      "description": "F = k|qQ| ÷ r² = |Q|E, the size of the force on the second charge.",
      "format": "number"
    },
    "interaction": {
      "label": "Force between the charges",
      "description": "Repulsive for charges of the same sign, attractive for opposite signs.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "field",
      "q": 2,
      "qu": "e",
      "r": "0.0265 nm",
      "E": 10000,
      "Eu": "NC",
      "q2": -1,
      "q2u": "e"
    },
    "outputs": {
      "result": "E = 4.101 × 10¹² N/C",
      "field": 4101002625466.093,
      "direction": "Away from the charge",
      "distance": 2.65e-11,
      "potential": 108.67656957485147,
      "force": 6.570530582494427e-7,
      "interaction": "Attractive: the charges pull together"
    },
    "text": "E = 4.101 × 10¹² N/C."
  },
  "examples": [
    {
      "given": {
        "find": "field",
        "q": 2,
        "qu": "e",
        "r": 2.65e-11
      },
      "expect": {
        "E": 4101002625466.093,
        "direction": "Away from the charge",
        "result": "E = 4.101 × 10¹² N/C"
      },
      "source": "OpenStax, University Physics Volume 2, §5.4 Electric Field (E = (1 ÷ 4πε₀) q ÷ r² for a point charge; F = QE, Equation 5.3; Example 5.3: the field of a helium nucleus (2e) at 26.5 × 10⁻¹² m is 4.1 × 10¹² N/C, pointing away), https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field (retrieved 2026-10-19); NIST, CODATA 2022: ε₀ = 8.8541878188(14) × 10⁻¹² F/m, https://physics.nist.gov/cgi-bin/cuu/Value?ep0; e = 1.602176634 × 10⁻¹⁹ C (exact), https://physics.nist.gov/cgi-bin/cuu/Value?e (retrieved 2026-10-19); hand calculation in content.mdx: E = 2 × 1.602176634 × 10⁻¹⁹ ÷ (4π × 8.8541878188 × 10⁻¹² × (26.5 × 10⁻¹²)²)"
    },
    {
      "given": {
        "find": "field",
        "q": 1,
        "qu": "e",
        "r": 5.29e-11,
        "q2": -1,
        "q2u": "e"
      },
      "expect": {
        "force": 8.244244234327191e-8,
        "interaction": "Attractive: the charges pull together"
      },
      "source": "OpenStax, University Physics Volume 2, §5.3 Coulomb’s Law (Equation 5.1, F = (1 ÷ 4πε₀)|q₁q₂| ÷ r²; Example 5.1: the electron and proton of hydrogen at 5.29 × 10⁻¹¹ m attract with 8.25 × 10⁻⁸ N), https://openstax.org/books/university-physics-volume-2/pages/5-3-coulombs-law (retrieved 2026-10-19) (8.25 × 10⁻⁸ N with k = 8.99 × 10⁹ and e = 1.602 × 10⁻¹⁹ C); NIST, CODATA 2022: ε₀ = 8.8541878188(14) × 10⁻¹² F/m, https://physics.nist.gov/cgi-bin/cuu/Value?ep0; e = 1.602176634 × 10⁻¹⁹ C (exact), https://physics.nist.gov/cgi-bin/cuu/Value?e (retrieved 2026-10-19); hand calculation in content.mdx"
    },
    {
      "given": {
        "find": "field",
        "q": 1,
        "qu": "nC",
        "r": 1
      },
      "expect": {
        "E": 8.9875517861708,
        "potential": 8.9875517861708,
        "result": "E = 8.98755 N/C"
      },
      "source": "OpenStax, University Physics Volume 2, §5.4 Electric Field (E = (1 ÷ 4πε₀) q ÷ r² for a point charge; F = QE, Equation 5.3; Example 5.3: the field of a helium nucleus (2e) at 26.5 × 10⁻¹² m is 4.1 × 10¹² N/C, pointing away), https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field (retrieved 2026-10-19); NIST, CODATA 2022: ε₀ = 8.8541878188(14) × 10⁻¹² F/m, https://physics.nist.gov/cgi-bin/cuu/Value?ep0; e = 1.602176634 × 10⁻¹⁹ C (exact), https://physics.nist.gov/cgi-bin/cuu/Value?e (retrieved 2026-10-19); hand calculation in content.mdx: k = 1 ÷ (4πε₀) = 8.98755 × 10⁹ N·m²/C², E = k × 10⁻⁹ ÷ 1²"
    },
    {
      "given": {
        "find": "distance",
        "q": 1,
        "qu": "uC",
        "E": 10000,
        "Eu": "NC"
      },
      "expect": {
        "r": 0.9480269925572161,
        "result": "r = 0.948027 m"
      },
      "source": "OpenStax, University Physics Volume 2, §5.4 Electric Field (E = (1 ÷ 4πε₀) q ÷ r² for a point charge; F = QE, Equation 5.3; Example 5.3: the field of a helium nucleus (2e) at 26.5 × 10⁻¹² m is 4.1 × 10¹² N/C, pointing away), https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field (retrieved 2026-10-19); hand calculation in content.mdx: r = √(k × 10⁻⁶ ÷ 10,000) = √0.898755"
    },
    {
      "given": {
        "find": "charge",
        "E": 10000,
        "Eu": "NC",
        "r": 1
      },
      "expect": {
        "charge": 0.0000011126500562018527
      },
      "source": "OpenStax, University Physics Volume 2, §5.4 Electric Field (E = (1 ÷ 4πε₀) q ÷ r² for a point charge; F = QE, Equation 5.3; Example 5.3: the field of a helium nucleus (2e) at 26.5 × 10⁻¹² m is 4.1 × 10¹² N/C, pointing away), https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field (retrieved 2026-10-19); hand calculation in content.mdx: |q| = 4πε₀ × 10,000 × 1² = 1.11265 × 10⁻⁶ C"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 2, §5.4 Electric Field: the field of a point charge E = (1 ÷ 4πε₀)(q ÷ r²) r̂; Equation 5.3, F = QE; Example 5.3 (the field of a helium nucleus, charge 2e, at 26.5 × 10⁻¹² m: 4.1 × 10¹² N/C, pointing away), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field",
    "OpenStax, University Physics Volume 2, §5.3 Coulomb’s Law: Equation 5.1, F = (1 ÷ 4πε₀)|q₁q₂| ÷ r²; Example 5.1 (the proton and electron of hydrogen at 5.29 × 10⁻¹¹ m: 8.25 × 10⁻⁸ N, attractive), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/5-3-coulombs-law",
    "OpenStax, University Physics Volume 2, §7.3 Calculations of Electric Potential: Equation 7.8, V = kq ÷ r for a point charge, with V = 0 at infinity, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/7-3-calculations-of-electric-potential",
    "NIST, CODATA 2022: vacuum electric permittivity ε₀ = 8.8541878188(14) × 10⁻¹² F/m, retrieved 2026-10-19. https://physics.nist.gov/cgi-bin/cuu/Value?ep0",
    "NIST, CODATA 2022: elementary charge e = 1.602176634 × 10⁻¹⁹ C (exact), retrieved 2026-10-19. https://physics.nist.gov/cgi-bin/cuu/Value?e"
  ],
  "related": [
    "ohms-law",
    "voltage-divider",
    "gravitational-force",
    "force"
  ],
  "changelog": []
}