How to use the calculator
- Enter any three of the four quantities: total charge Q, radius R, axial distance z, and field magnitude E.
- Leave the unknown quantity blank (completely empty).
- Select Compute Missing Quantity to solve for the blank field.
Input rules: the calculator expects nonnegative values for most cases because it reports field magnitude.
If you need a signed direction (positive/negative axis), compute the magnitude here and then apply the sign based on your
coordinate convention and whether the charge is positive or negative.
Quick tip for reliable results: use SI units (coulombs and meters) and avoid mixing centimeters with meters.
If your problem statement uses cm or μC, convert first or use scientific notation (for example, 10 cm = 0.10 m and
2 μC = 2e-6 C).
The magnitude of the electric field on the axis of a uniformly charged ring is:
Formula: E = (k Q z) / z^2+R^2^3/2
- = electric field magnitude on the axis (N/C)
- = total ring charge (C)
- = ring radius (m)
- = axial distance from the center (m)
- = Coulomb constant ≈ 8.9875517923×109 N·m²/C²
This calculator uses the same relationship in reverse when you leave one field blank. Solving for
or
is algebraic.
Solving for is more subtle because
appears both in the numerator and inside the power in the denominator.
The script therefore uses an iterative numerical method (a Newton-style update) to converge on a distance that matches the
requested field.
Worked example (step-by-step)
Consider a ring with a modest amount of charge and a radius typical of a lab demonstration. We will compute the axial field
at a point closer than one radius away.
- Ring charge: Q = 2 μC =
2e-6 C
- Ring radius: R = 0.10 m (10 cm)
- Axial distance: z = 0.05 m (5 cm)
Steps:
- Enter
2e-6 into Ring Charge Q.
- Enter
0.10 into Ring Radius R.
- Enter
0.05 into Axis Distance z.
- Leave Electric Field E blank and click Compute Missing Quantity.
The calculator returns a field magnitude of roughly 287 N/C.
If you change only the distance to z = 0.10 m, the field drops to about 112 N/C.
That decrease is expected: as you move away, the denominator grows quickly because it contains
.
Direction note: this tool reports magnitude only. For a positive charge, the field points away from the ring
along the +z direction if you define +z to be on the observation side. For a negative charge, the direction reverses.
If your problem uses a signed coordinate (±z), apply the sign after computing the magnitude.
Assumptions and limitations
- On-axis only: The formula used here is valid only for points on the ring’s symmetry axis. Off-axis fields require a different integral or a numerical field computation.
- Uniform charge distribution: The ring is assumed to be thin and uniformly charged. If charge density varies with angle, the symmetry cancellation is incomplete and the axial-only result changes.
- Electrostatic approximation: Charges are treated as stationary. The model ignores radiation, time-varying magnetic fields, and material polarization effects.
- Magnitude-only output: The computed is nonnegative. Assign direction separately.
- Edge case at z = 0: At the exact center of the ring, the axial field is zero for any finite charge. That means you cannot infer a unique charge from a nonzero field at .
- Solving for z is numerical: Some combinations of inputs can be ill-conditioned (for example, extremely small fields with very large radii). If the solver struggles, try values closer to the expected scale or verify unit conversions.
Interpretation tips and sanity checks
The ring’s axial field has several useful behaviors that help you check whether a result “makes sense”:
-
Near the center
():
the field grows approximately linearly with .
This is why the field is exactly zero at the center and increases as you move away.
-
At large distances
():
the ring behaves like a point charge, and the field approaches
.
If you are many radii away, your answer should be close to the point-charge value.
-
Maximum along the axis: for a fixed and , the axial field magnitude is not largest at the center.
It increases from zero, reaches a maximum around , and then decreases.
This is a common conceptual checkpoint in electrostatics.
Another practical check is dimensional analysis: the expression contains (units N·m²/C) multiplied by a length and divided by a length cubed, leaving N/C.
If you accidentally enter centimeters as meters, the field can be off by factors of 100 or 10,000.
Units and conversions (common classroom values)
Many problems use small charges and small rings. The calculator accepts SI units directly, but these conversions help you
translate typical textbook numbers:
- 1 μC (microcoulomb) =
1e-6 C
- 1 nC (nanocoulomb) =
1e-9 C
- 1 cm =
0.01 m
- 10 cm =
0.10 m
- 1 mm =
0.001 m
Scientific notation is supported in all fields. For example, if a ring has 50 nC of charge and a radius of 2.5 cm,
you can enter 5e-8 for Q and 0.025 for R.
This is often faster and less error-prone than typing many zeros.
FAQ
Does the calculator handle negative charge?
The input fields are configured for nonnegative values because the output is a magnitude. In physics, a negative charge
reverses the field direction, not the magnitude. If you need a signed answer, compute the magnitude here and then apply a
negative sign according to your coordinate system.
Introduction: Why can’t I solve for Q when z = 0 and E is nonzero?
At the center of a uniformly charged ring, symmetry forces the axial field to be exactly zero. No finite charge can produce
a nonzero axial field at that point. The calculator detects this and shows an explanatory message.
Why might solving for z fail to converge?
When solving for , the script uses an iterative method that
relies on the slope (derivative) of the field function. If the derivative becomes extremely small or the requested field is
inconsistent with the other parameters, the update step can become unstable. In practice, this is usually caused by unit
mistakes (cm vs m) or by choosing values that imply an unrealistically tiny field at a very small distance.
Is this the same as the field of a disk?
No. A uniformly charged disk has a different axial field formula because charge fills the area, not just the circumference.
A ring is a one-dimensional distribution (a loop). A disk can be built by integrating rings of different radii, but the final
expression is different.
What if I know linear charge density instead of total charge?
If you have linear charge density (C/m), convert to total charge using
.
Then enter the resulting into the calculator.
Representative values (for intuition)
The table below lists a few representative values for a fixed charge and radius. It is included to help you build intuition
about how the field changes with axial distance. You can reproduce these numbers by entering Q and R and then solving for E
at each z.
All calculations run locally in your browser using JavaScript. No inputs are sent to a server, and nothing is stored unless
your browser itself stores form values. You can bookmark this page or save it for offline study.
If you are using the calculator in a lab setting, this local-only behavior is useful when internet access is limited.
If you are writing up results, consider copying the computed summary using the Copy Summary button. It formats
the values in a consistent way so you can paste them into a lab notebook, homework solution, or design notes.
