Converting Candelas to Lumens for Car Headlights

[YoshiMoshi]

Homework Statement

Convert 15,000 to 20,000 candela to lumens

Relevant Equations

Candella = lumen/( (2 * pi) (1 - cos( 0.5*( apex angle ) ) )

I found the following law for the state of CT

1. Type 2 or 2A Lights—Upper beam limited to 20,000 to 75,000 candela per lamp. Lower beam limited to 15,000 to 20,000 candela per lamp.
2. Type 1 or 1A Lights—Upper beam limited to 18,000 to 60,000 candela per lamp.

I can't find a definition of Type 2 or 2A vs Type 1 or 1A.

I'm trying to make sure that the headlights that I got are road legal. So headlight brightness is measured in lumens when you go and purchase headlight bulbs. So if a bulb is 6,000 lumnes, are they compliant with the above law? The issue is that the law is measured in candela but car bulb manufactures specify brightness of their bulbs in lumens.

I've been trying to convert the two measurements but am having great difficulty.

Candella = lumen/( (2 * pi) (1 - cos( 0.5*( apex angle ) ) )

 

[Cutter Ketch]

Is that the whole homework statement? You can’t just convert from one to the other. They aren’t different units of the same thing. Lumen refers to the total amount of light being emitted (weighted according to the sensitivity of the eye). Candela refers to the amount of light being emitted within some cone angle (also weighted by the sensitivity of the eye and normalized to 1 steradian) pointed in some direction. To determine candela from lumens they would have to say what the angular distribution is and in which direction they want to know.

 

[Cutter Ketch]

Interesting law. Why would a limit have a range?

How bright the brightest part of the beam is depends on how the beams are distributed in angle space. Your cone angle formula assumes that, at a large distance from the lamp (“large” compared to the size of the lamp), the light is uniformly distributed over a circle. This is never the case with headlights. I doubt you will ever learn enough about the angular distribution to accurately compute candela. You either have to measure, or since the law you quote is federal law, assume that headlamps sold in the US are compliant.

 

[YoshiMoshi]

This is a real life problem, that I'm trying to answer. I'm not sure why the limit gives a range, that doesn't make sense to me either. I have never seen candela specified on headlight bulbs, always lumen. So I'm not sure how to solve the problem. It is strange that the federal law limits the brightness in candela, but light bulb manufactures do not measure the brightness in candela, but instead lumens. There doesn't seem to be a way to convert the two.

Amazon product page: BUNKER INDUST 9006/HB4 LED Headlight Bulbs, X3 Series 12000LM Super Bright Headlights Conversion Kit, 6000K Cool White IP67 Waterproof

https://www.amazon.com/gp/product/B07VHM4Z1Q/?tag=pfamazon01-20

The candela is not specified, and I'm not able to find a datasheet. Is there a tool to measure candela, and I can hold up the tool the headlight bulb with the light on and get a accurate measurement of candela?

 

[haruspex]

Is there a tool to measure candela, and I can hold up the tool the headlight bulb with the light on and get a accurate measurement of candela?

You could illuminate a white wall and guesstimate the angle of principal illumination. It,s a bit rough because the central area will be brighter just because it is nearer.

 

[Cutter Ketch]

I don’t really see a problem with manufacturers specifying lumens and laws specifying candela. That makes sense to me. They are for different purposes. Ok, so you can’t convert (without some info you don’t have). The government wouldn’t check the brightness by converting the manufacturer’s specification. They would put a light meter in the brightest part of the beam and measure the intensity.

 

[YoshiMoshi]

I don’t really see a problem with manufacturers specifying lumens and laws specifying candela. That makes sense to me. They are for different purposes. Ok, so you can’t convert (without some info you don’t have). The government wouldn’t check the brightness by converting the manufacturer’s specification. They would put a light meter in the brightest part of the beam and measure the intensity.

Wouldn't the light meter specify the "Lux" and not candela or lumens? Is there a way to convert lux to candela or lumens?

 

[Cutter Ketch]

Is there a way to convert lux to candela or lumens?

Yes. Lux is the unit of Illuminance defined as luminous flux per unit area in units of lumens per square meters. Candela is the unit of luminous intensity defined as luminous flux per solid angle in units of lumens per steradian. If you measure Illuminance E in lux at a distance R in meters from the source, then the luminous intensity I in candela is

I = E / R^2

 

[YoshiMoshi]

So there appears to be no way to determine if the headlight bulbs you purchase for your car are compliant with the federal law, unless you go and buy them and take a measurement? This seems to be highly flawed. The law defines the amount of candela that are permitted. However manufactures sell light bulbs with a specified lumen value and not a specified candela value.

This is illogical.

However I did find this plot. So the maximum Illuminance is 50 lx. What value do I take for R?

It seems that I could take any value within the red boundary. Obviously closer to the light source would give me a higher candela value because of the relationship, candela = lux / distance^2.

So If i take the maximum distance of say approximately 60 meters

(50 lx)/(60 m)^2 = 0.01388888888 candela

Am I doing something wrong here? This seems really small and no where close to the federal limit. If I take 1 meter, I get 50 candela.

I'm a bit confused here on how do I select the distance, because based on the difference I select I will get a difference candela value.

 

[Cutter Ketch]

These kind of laws are not usually enforced on the consumer. The manufacturers are responsible for compliance. If they are marketed in the US through a reputable channel, they are probably compliant.

 

[Cutter Ketch]

What value do I take for R?

On the graph you will find that for some straight line from the source the Illuminance (in lux) falls with distance proportional to 1/R^2. You will get the same luminous intensity (in candela) no matter what R you choose. (caveat: depending on how they overlapped the pair of headlights). That’s why luminous intensity is a useful quantity. It is a fixed property of the beam that doesn’t change with distance.

first I must apologize. I got the relation backwards. It’s not

I = E / R^2

it’s

I = E*R^2

I swapped E and I. Sorry.

the graph is a little hard to read, plus it isn’t clear where within each color band the value occurs. However in the top graph the peak appears to be something like 50 lx at 45 m = 100,000 cd, and 25 lx at 65 m = 100,000 cd and 10 lx at 100 m = 100,000 cd. I have to admit I’m having trouble reconciling the remaining color bands. The luminous intensity seems to be growing significantly increasing to 500,000 cd, which can’t happen. I suspect some looseness in the graph for marketing purposes.

Note that even at 100,000 cd the lamp is probably still legal because the diagram to the left suggests that the graph is for a pair of headlights and they probably overlapped them to get the most impressive graph.

 

[YoshiMoshi]

Why do you say the peak for 50 lx is at 45 meters for the top graph? When you say peak, you mean the furthest distance at which the red region is? To me it looks like a little more than 50 meters.

25 lx for the top graph is hard to tell, somewhere between 50 meters and 100 meters

10 lx for the top graph is hard to tell, somewhere between 100 meters and 200 meters.

The number I got appear to not be close to the numbers you got.

I just want to make sure, what is meant by peak?

Is there a way to get the values for a single bulb?

 

[Cutter Ketch]

Why do you say the peak for 50 lx is at 45 meters for the top graph? When you say peak, you mean the furthest distance at which the red region is? To me it looks like a little more than 50 meters.

25 lx for the top graph is hard to tell, somewhere between 50 meters and 100 meters

10 lx for the top graph is hard to tell, somewhere between 100 meters and 200 meters.

The number I got appear to not be close to the numbers you got.

I just want to make sure, what is meant by peak?

Is there a way to get the values for a single bulb?

I meant peak in angle, the line along the peaks. As to where in the color band the stated value is realized, you can’t really tell. It could be at the boundary between the colors, but I was guessing it was more in the middle of the color band. That seemed to make more sense for 1/R^2. However it’s more of a crap shoot than an exact analysis, more of an example of the idea than anything you would want to trust (particularly seeing as the luminous flux grows at very large distance. Highly dubious)