Newton's rings lens radius of curvature
Turn one measured Newton's ring into the lens curvature: a dark ring of radius 0.0025 m at order 5 under 589 nm light means the lens surface has a radius of curvature just over 2 m. Free in your browser.
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Worked example
- Ring radius
- 0.0025
- Order
- 5
- Wavelength
- 0.000000589
- Ring type
- dark
- Radius of curvature
- 2.1222410866
- Precision
- 12
Newton's rings appear when a plano-convex lens rests on an optically flat plate and monochromatic light reflects from the air film between them. The circular interference fringes encode the lens radius of curvature R: once you measure the radius r of the m-th dark or bright ring and know the wavelength λ, a short closed-form formula recovers R without a full interferometer model. This Newton's rings lens radius of curvature calculator accepts the ring radius, the integer order, and the wavelength, applies the standard reflected-light dark-ring relation R equals r squared over m λ (or the bright-ring form when you select bright fringes), and returns the curvature radius with a formula string you can show next to lab notes. Units stay consistent: if r and λ are in metres, R is in metres; if you work in millimetres throughout, R is in millimetres.
What Newton's rings measure and why R equals r squared over m lambda
When a convex glass surface of radius of curvature R contacts a flat plate, the air gap thickness at radial distance r from the contact point is, to an excellent approximation for r much smaller than R, t equals r squared divided by two R.
In reflected monochromatic light of wavelength λ there is a half-wave phase shift at the denser bottom surface, so the condition for destructive interference (dark rings) is two t equals m λ with integer order m equals one, two, three, and so on counting outward from the central dark spot. Substituting the gap formula immediately gives r sub m squared equals m λ R, which rearranges to the working laboratory expression R equals r squared over m λ.
Bright rings obey two t equals half an odd multiple of λ, leading to R equals two r squared over open parenthesis two m minus one close parenthesis λ. Measuring a ring diameter D instead of a radius is common: use r equals D over two, or equivalently R equals D squared over four m λ for dark rings. The geometry assumes a spherical lens surface, normal incidence, and an air film of refractive index one; it does not correct for finite beam angle, contact deformation, or immersion liquids.
What you can do with it
Undergraduate Newton's rings lab checks
Convert travelling-microscope ring radii, order m, and laser or sodium wavelength into lens radius of curvature R for lab reports and auto-graded homework.
Notebook and instrument automation
Script single-ring curvature estimates from measured fringe radii without importing a full optics library; reject invalid orders before they corrupt a fit.
FAQ
Should I use ring radius or diameter?
This calculator expects the ring radius r. If you measured diameter D, pass r = D/2. Dark rings then match the textbook form R = D²/(4mλ).
What formula links a dark ring to the lens curvature?
For dark rings in reflected light, R equals the ring radius squared divided by the order times the wavelength. It comes from the thin air gap between the curved lens and the flat plate.
Can I use bright rings instead of dark ones?
Yes. Choose bright rings and the calculation switches to R equals two times the ring radius squared, divided by twice the order minus one, times the wavelength. Dark rings are used when you choose nothing.
Which units should the ring radius and wavelength be in?
Use the same length unit for both. If both are in metres, the curvature comes out in metres; if you work in millimetres throughout, it comes out in millimetres.
Can it fit several rings at once?
No. It works out the curvature from a single ring measurement. For a fit over several rings, plot the squared radius against the order yourself and use the slope.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/optics/newtons-rings-lens-radius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"ring_radius":0.0025,"order":5,"wavelength":5.89e-7}'const res = await fetch("https://api.kit.forhosting.com/optics/newtons-rings-lens-radius", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"ring_radius": 0.0025,
"order": 5,
"wavelength": 5.89e-7
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/newtons-rings-lens-radius",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"ring_radius": 0.0025,
"order": 5,
"wavelength": 5.89e-7
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/newtons-rings-lens-radius", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"ring_radius":0.0025,"order":5,"wavelength":5.89e-7}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"ring_radius":0.0025,"order":5,"wavelength":5.89e-7}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/newtons-rings-lens-radius", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"ring_radius": 0.0025,
"order": 5,
"wavelength": 5.89e-7
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.newtons_rings_lens_radius",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Invalid orders, non-positive lengths, and unknown ring types fail with invalid_input and are not billed.
What Newton's rings measure and why R equals r squared over m lambda
This endpoint takes the radius form so you can feed travelling-microscope half-diameter readings directly.
Input contract, response shape, and error behaviour
Optional precision is an integer from zero through fifteen controlling decimal places on numeric outputs, defaulting to twelve. Keep r and λ in the same length unit so R is self-consistent. Pair the response with your own plots of r squared versus m if you want a least-squares multi-ring fit; this capability evaluates a single ring measurement in closed form and does not fit slopes or emit images. POST a JSON body with three required fields: ring_radius (positive finite number r, aliases r, radius, ringRadius), order (positive integer m greater than or equal to one, aliases m, n, ring_order, ringOrder), and wavelength (positive finite number λ, aliases lambda, λ, wl). Optional ring_type is the string dark or bright and defaults to dark for the most common reflected-light dark-ring lab series; synonyms such as black or light are accepted and normalized. Missing fields, non-finite values, zero or negative lengths, fractional or non-positive order, and unknown ring_type values fail with invalid_input naming the problem; those failures do not consume prepaid balance. On success the body echoes ring_radius, order, wavelength, and the normalized ring_type; returns radius_of_curvature as the computed R; includes precision; and provides formula and method strings for display and logging. Example: ring_radius equals zero point zero zero two five metres, order equals five, wavelength equals five point eight nine times ten to the minus seven metres (sodium D light) with dark rings yields R equals r squared over five λ, a radius of curvature of order a few metres typical of a long-focus plano-convex lens.
Where a Newton's rings radius-of-curvature API earns its keep
Undergraduate optics labs still grade Newton's rings experiments by whether students recover a plausible lens R from travelling-microscope readings of successive dark rings. An automated checker needs a deterministic oracle that implements R equals r squared over m λ without spreadsheet drift and that rejects m equals zero or negative wavelengths instead of dividing by zero silently. Instrument firmware and notebook pipelines that already measure fringe radii can call a tiny parse API instead of re-deriving bright versus dark cases under deadline pressure. Use one ring for a quick lab check, or loop orders externally when you want a multi-ring average—this endpoint stays intentionally small so edge workers and LMS plugins can depend on it without standing up a full optical design suite.
Integration use cases
Curriculum and SEO live widgets
Embed a free browser calculator that matches the API formula R = r²/(mλ) so students and search visitors get the same dark-ring or bright-ring result.
Technical questions
How do I compute lens radius of curvature from Newton's rings?
Send positive ring_radius r, integer order m ≥ 1, and positive wavelength λ. For dark rings (default) the API returns R = r²/(mλ). Set ring_type to bright for R = 2r²/((2m−1)λ).
What units should ring_radius and wavelength use?
Any consistent length unit. Metres throughout yield R in metres; millimetres throughout yield R in millimetres. Do not mix units.
Why is order m required to be a positive integer?
Ring order indexes successive dark or bright fringes. m = 0 is the contact point, not a usable radius measurement, and non-integer m is outside the standard discrete fringe model.
What does the Newton's rings lens radius calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.