Tools in This Collection
Lens Equation Calculator
Solve 1/f = 1/dₒ + 1/dᵢ for optics problems
Electric Field Calculator
Calculate electric field strength E = F/q at a point
Capacitance Calculator
Calculate capacitance, charge, and stored energy
Coulomb's Law Calculator
Calculate electrostatic force between charges: F = kq₁q₂/r²
Electrostatics and Optics Calculations
Electrostatics and optics problems both rely on inverse-square relationships and follow a similar problem-solving pattern: identify the source, calculate the field at a point, then find the effect on a charge or ray. The core equations are Coulomb's law for electrostatic forces, E = F/q for electric fields, and the thin lens equation for optics.
Coulomb's Law: F = kq₁q₂/r²
Coulomb's law gives the electrostatic force between two point charges: F = kq₁q₂/r², where k = 8.99 × 10⁹ N·m²/C². Two charges of 1 μC (1 × 10⁻⁶ C) separated by 0.1 m experience a force of F = 8.99 × 10⁹ × (10⁻⁶)² / (0.1)² = 8.99 × 10⁹ × 10⁻¹² / 0.01 = 0.899 N. The force is repulsive for like charges and attractive for opposite charges. The Coulomb's Law Calculator solves for force, either charge, or separation distance.
Electric Field: E = F/q
The electric field at a point is the force per unit positive test charge: E = F/q, measured in N/C (or V/m). For a point charge Q, the field at distance r is E = kQ/r². For Q = 5 μC at r = 0.5 m: E = 8.99 × 10⁹ × 5 × 10⁻⁶ / 0.25 = 179,800 N/C ≈ 180 kN/C. Field lines point away from positive charges and toward negative charges. The Electric Field Calculator computes field strength and direction for point charge configurations.
Capacitance: C = Q/V
Capacitance C = Q/V is the ratio of stored charge (Q in Coulombs) to voltage (V in volts), measured in Farads. For a parallel-plate capacitor: C = ε₀A/d, where ε₀ = 8.85 × 10⁻¹² F/m is the permittivity of free space. Capacitors in parallel add directly (C_total = C₁ + C₂), while capacitors in series combine as 1/C_total = 1/C₁ + 1/C₂. The energy stored in a capacitor is U = ½CV². The Capacitance Calculator handles charge, voltage, and energy calculations for single and combined capacitors.
Thin Lens Equation: 1/f = 1/dₒ + 1/dᵢ
The thin lens equation connects focal length (f), object distance (dₒ), and image distance (dᵢ): 1/f = 1/dₒ + 1/dᵢ. For a converging lens with f = 20 cm and an object placed at dₒ = 30 cm: 1/dᵢ = 1/20 − 1/30 = 3/60 − 2/60 = 1/60, so dᵢ = 60 cm. The image is real and inverted. Magnification is m = −dᵢ/dₒ = −60/30 = −2 (twice the size, inverted). If the object is inside the focal length, the image is virtual, upright, and magnified — this is how a magnifying glass works. The Lens Equation Calculator handles all combinations of focal length, object and image distances, and magnification.
Frequently Asked Questions
What is Coulomb's law and when do I use it?
Coulomb's law F = kq₁q₂/r² gives the force between two stationary point charges. Use it when you need the electrostatic force between two charged particles or objects separated by a known distance. k = 8.99 × 10⁹ N·m²/C². The force is attractive for opposite charges and repulsive for like charges. For moving charges or magnetic effects, you need additional laws from electromagnetism.
How do I use the thin lens equation to find image location?
Use 1/f = 1/dₒ + 1/dᵢ. Rearrange to 1/dᵢ = 1/f − 1/dₒ, then take the reciprocal to get dᵢ. Positive dᵢ means a real image on the other side of the lens; negative dᵢ means a virtual image on the same side as the object. For f = 20 cm, dₒ = 30 cm: 1/dᵢ = 1/20 − 1/30 = 1/60, so dᵢ = 60 cm (real, inverted image).
How do capacitors combine in series vs parallel?
Capacitors in parallel add directly: C_total = C₁ + C₂ + ... — they share the same voltage but accumulate more total charge. Capacitors in series combine as 1/C_total = 1/C₁ + 1/C₂ + ... — total capacitance is less than the smallest individual capacitor. Two identical 10 μF capacitors in parallel give 20 μF; in series, they give 5 μF.