Electric Field Calculator

The Electric Field Calculator computes the electric field strength produced by one or many point charges, using \( E = k_{e}\,q / (\varepsilon_{r}\, r^{2}) \) for a single source and full vector superposition \( \vec{E}_{\text{net}} = \sum_{i} \vec{E}_{i} \) for multi-charge problems. Switch between Single-charge mode (solve for E, q, or r in one form) and Multi-charge mode (place up to six charges anywhere in the 2D plane and read the net field at any test point P), enter charges in coulombs, microcoulombs, nanocoulombs, or elementary charges e, and the calculator returns the field magnitude in V/m and N/C, the components Eₓ and Eᵧ, the direction angle θ, the electric potential V at the test point, the force on a 1 µC probe charge, and a step-by-step LaTeX derivation. A live SVG redraws the charge orbs (red for +, blue for −), the per-charge contribution arrows, and the resulting net field vector as you type.

How to Use This Electric Field Calculator

1. Pick a mode at the top. Single point charge uses the closed-form \( E = kq/r^{2} \). Multi-charge superposition lets you place up to six charges in the 2D plane and reads the net vector field at any chosen test point.
2. In single mode, choose what to solve for (E, q, or r) — the matching input hides itself automatically so you cannot accidentally over-constrain the problem. Type the remaining two quantities with their preferred units.
3. In multi mode, fill one row per source charge (value + unit + x + y). Leave a row blank to skip it. Then enter the test-point coordinates (x, y) and the shared coordinate unit.
4. Choose the surrounding medium. Vacuum and air leave the field unchanged. Water at εᵣ ≈ 80 screens the field by roughly two orders of magnitude. Pick Custom εᵣ for an unusual dielectric.
5. Press Calculate and read the field magnitude, direction, per-charge contributions, the step-by-step derivation, and the animated field-line / superposition diagram.

What Makes This Calculator Different

The Formula in One Line

For a single point charge of value \( q \) at distance \( r \) inside a medium of relative permittivity \( \varepsilon_{r} \), the electric field magnitude is

\[ E \;=\; k_{e}\,\dfrac{q}{\varepsilon_{r}\,r^{2}} \]

where \( k_{e} = 1/(4\pi\varepsilon_{0}) \approx 8.9875 \times 10^{9}\) N·m²/C² is Coulomb's constant. The field is a vector that points radially outward from a positive source charge and radially inward toward a negative one — that is, in the direction a positive test charge would be pushed (or pulled).

For many charges, by the principle of superposition the net field at any point is the vector sum of the individual contributions:

\[ \vec{E}_{\text{net}}(\vec{r}) \;=\; \sum_{i} k_{e}\,\dfrac{q_{i}}{\varepsilon_{r}\,|\vec{r}-\vec{r}_{i}|^{2}}\,\hat{r}_{i} \]

The calculator computes each \( \vec{E}_{i} \) separately, decomposes it into Eₓ and Eᵧ components, sums those component-wise, then reconstructs the magnitude \(|E| = \sqrt{E_{x}^{2}+E_{y}^{2}}\) and the direction \( \theta = \arctan(E_{y}/E_{x}) \).

Worked Example: 1 µC at 10 cm

• \( E = (8.9875 \times 10^{9}) \times (1 \times 10^{-6}) / (0.10)^{2} \approx 8.99 \times 10^{5}\) V/m — about 900 kV/m.
• The field points outward from the positive charge. A free electron placed there would feel a force \( F = qE \approx 1.44 \times 10^{-13}\) N pointing toward the source.
• Electric potential at this distance: \( V = kq/r \approx 89.9\) kV — explaining why even a small static-charged conductor can give you a noticeable jolt.

Worked Example: Electric Dipole

Place \(+1\) µC at \((-2\) cm, 0) and \(-1\) µC at \((+2\) cm, 0). The test point sits at the dipole's midpoint \((0, 1\) cm)\) just above the axis.

• Distance from each charge to P: \( r = \sqrt{2^{2}+1^{2}}\) cm \(= \sqrt{5}\) cm ≈ 2.24 cm.
• Each contribution has magnitude \( |E_{i}| = kq/r^{2} \approx 1.8 \times 10^{7}\) V/m.
• The y-components cancel by symmetry; the x-components add up along the −x direction (toward the negative charge). The net field points horizontally with magnitude roughly \( 2 \times |E_{i}| \cos\theta \) where \(\cos\theta = 2/\sqrt{5}\).
• This is the iconic "dipole field" that you'll meet again every time you study polar molecules, antennas, or NMR.

Electric Field vs Electric Force vs Electric Potential

These three quantities describe related but distinct things:

• Electric field \(\vec{E}\) (V/m or N/C) — the force per unit positive test charge at a point. Exists even if no test charge is present. Vector.
• Electric force \(\vec{F} = q\vec{E}\) (newtons) — what actually happens to a charge \(q\) when you place it in the field. Vector.
• Electric potential \(V\) (volts) — the work per unit positive test charge to bring a charge from infinity to that point. Scalar. Its negative gradient is the electric field: \(\vec{E} = -\nabla V\).

The calculator returns all three so you can cross-check your understanding.

Common Electric-Field Magnitudes

Tips for Multi-Charge Problems

• Use symmetry first. If charges sit symmetrically about the test point, some components cancel exactly. The calculator confirms this — you'll see Eₓ or Eᵧ come out to (very close to) zero.
• Pick the test point carefully. Choosing it on an axis of symmetry simplifies the math (and lets you sanity-check the calculator output).
• Watch the signs. A positive contribution arrow points from the source toward the test point. A negative one points from the test point toward the source. Mixing them up flips the net direction by 180°.
• Coordinate unit is shared. All six charges and the test point use the same coordinate unit you chose at the bottom of the multi-charge section. This keeps the geometry consistent.

Frequently Asked Questions

What is the formula for the electric field of a point charge?
\( E = k_{e}\,q / r^{2} \) where \(k_{e} \approx 8.9875 \times 10^{9}\) N·m²/C². The field points outward from a positive charge and inward toward a negative one.

What are the units of electric field?
SI: V/m (volts per meter), equivalent to N/C (newtons per coulomb). The calculator accepts both and converts internally.

How do I add fields from multiple charges?
Use vector superposition: compute each charge's contribution as a 2D vector, sum the x-components and y-components separately, then reconstruct the magnitude as \(\sqrt{E_{x}^{2}+E_{y}^{2}}\) and the direction as \(\arctan(E_{y}/E_{x})\). The multi-charge mode of this calculator automates exactly that.

What is the difference between electric field and electric force?
The field describes the influence of a source charge on the surrounding space. The force \( F = qE \) is what happens when you place another charge \(q\) in that field. The field exists everywhere; the force only acts on charges that are actually present.

Does the medium between source charges change the field?
Yes. The field is divided by the relative permittivity εᵣ of the medium. Air ≈ 1, water ≈ 80. The same source charge produces a field in water that is about 80× weaker than in vacuum — which is why ionic salts dissolve so readily in water.

What is the dielectric breakdown field of air?
About 3 × 10⁶ V/m (3 MV/m) for dry air at sea level. Above this, air ionizes and the geometry sparks. The calculator flags any result above this threshold.

Can I solve for the source charge or the distance?
Yes — in single-charge mode use the Solve for selector. The calculator rearranges \( E = kq/r^{2} \) into the matching closed form (\( q = E\varepsilon_{r}r^{2}/k \) or \( r = \sqrt{kq/(\varepsilon_{r}E)} \)) and hides the unknown input.

Why does my net field come out to zero?
Two equal but opposite charges placed at mirror positions relative to your test point produce equal-and-opposite contributions that exactly cancel — the field on the perpendicular bisector midpoint of a dipole is zero on the axis through the dipole. This is real physics, not a calculator error. Move the test point off the symmetry plane to see a non-zero field.