Solar farms don't belong in Norway

Classical Mechanics Level 1

The highest intensity of sunlight is only available on Earth’s equator. At any other latitude, the incident sunlight is spread over a larger area, decreasing the intensity of the sunlight.

The amount of solar energy collected by a single solar panel can be maximized by tilting the panel. For example, to collect the greatest amount of energy with a single solar panel in Greece at $40^{\circ} \textrm{N}$ latitude, you should angle the panel at $40^{\circ}$ .

Suppose you are building a solar farm on level ground to power a town in Greece. To decrease your environmental impact, you would like to minimize the square-shaped land area affected by the the solar farm – no one wants a row of unsightly panels around the town or a shadow cast on their backyard. Your solar farm needs employ as many commercially available solar panels as necessary to power the nearby town.

How much land area can be saved by tilting the solar panels at $40^{\circ}$ with one edge on the ground vs. laying them flat?

20% of land area could be saved 33% of land area could be saved 50% of land area could be saved No land area can be saved

8 solutions

Mark Angelo Valdejueza
Nov 11, 2018

Tilting vs. Laying Flat

The problem with tilting the solar panels are the shadow casted by the one on front. Suppose there are only 2 solar panels. Tilting them 40 degrees perpendicular to the sun will create shadow 90 degrees from the top to the ground. If we are talking "100%" efficiency, you have to allot space for the shadow. Let say the solar panel is 1 meter slanting height, this will create an approximate shadow of 1.3m which is where the second panel should go to also get "100%" efficiency. If you lay them flat, there is no shadow and the panels could be connected.

But what about if you could elevate them on a slanted plane? The solar panels layered behind the first layer would be positioned higher than the first layer, giving room for more land area. https://goo.gl/images/Uz6a3o shows an example of how I mean

Ruth Mckinlay - 2 years, 7 months ago

+1 for this. The question asks about saving land area. Casting a shadow over the neighbours wasn't given as a constraint. The most efficient use of land area is to build a large rectangular ramp of panels tilted at 40 degrees. You reduce the direct land footprint by a factor of cos(40) or about 0.766.

That said you also increase construction costs significantly and annoy the neighbours and/or create a vegetation 'dead zone' in the shadow of your massive ramp...

Conor McMenamin - 2 years, 7 months ago

No, the most effective use of land area is to stack them into 1 giant vertical pillar, therefore only using the land area of 1 panel:

Adam Nicholas - 2 years, 6 months ago

1 This is precisely what i was thinking about and calculating. The question is not specific enough to dismiss this kind of layout.

Dinko Koikov - 2 years, 7 months ago

Similarly, the panels could be placed in one long horizontal row (impractical, but possible).

William Allbritain - 2 years, 7 months ago

+1 I agree.The question doesn't have enough necessary details.

Тимур Сулейменов - 2 years, 7 months ago

Too many physicists and not enough engineers here. The saving is theoretically only on the edge furthest North, so the saving depends on the number of rows. Maximum saving about 23% with one square panel. But as the panels will have a frame which can be in shadow without loss, you could nudge them closer. Also as the sun hits them at different times of day.... so the shadow is only a problem at midday. What is the value of saved land versus efficiency lost? Etc.

James Maybury - 2 years, 7 months ago

What if, instead of placing individual panels on the ground at a 40 degree to the sun, one were to fuse all of the panels onto a single plane (effectively making one giant solar panel) and position the entire plane at a 40 degree angle to the sun? That would reduce the required land footprint by 20%.

Izosa Ize - 2 years, 7 months ago

Exactly my thought.... Poorly written question....

Gene Waldenmaier - 2 years, 7 months ago

The question is vague; it's not defined what "save" means. Is it acceptable to "save" on your own land use by casting a shadow on your neighbor's parcel?

Tapani Lindgren - 2 years, 7 months ago

As no engineering constraints were given we can put a panel of infinite height perpendicular to the earth and save 100% of the space.

Alex Simkin - 2 years, 6 months ago

Thumbs DOWN!! Your answer is correct according to physics. However, it is not correct to the question because the question itself is not clear. Nothing prevents me from joining the panels into a big panel and then tilt it! Or if you like, consider the panels are placed in one single row and then tilted about the axis which is the longer dimension. In this situation, it's a certainty that we can save more than 20% of the land. In other words, the question is NOT brilliant

庭豐石 - 2 years, 6 months ago

You then have a shadow under your giant panel monstrosity, making it not usable for growing (which I assume they are asking about)

George Ritz - 2 years, 6 months ago

AGREED TO YOU !

Monika Dixit - 2 years, 6 months ago

Why does the shadow casted 'cost' land area? Specifically the shadow cast by the rearmost panels.

Willem Monsuwe - 2 years, 6 months ago

Richard Desper
Nov 12, 2018

So...what's to stop me from having one solar panel placed at a 40 degree angle above my farm? According to the diagram, (a) the solar panel would be more efficient, producing more energy per square foot of the panel, and (b) it would have a smaller footprint. Yes, it would create a shadow, but if the shadow falls on my neighbor's land, why do I care?

I choose the "no land area can be saved" choice because the question did not offer a choice sufficiently close to (1 - cos(40)).

Yes, the question does not say that each solar panel has to have an edge on the ground. There's nothing stopping us from building some scaffolding and letting the solar panels reach into the sky. That would take approximately 35.8% less land, which is why 33% is a tempting answer.

David Henderson - 2 years, 7 months ago

Isnt it even (1-cos(40)^2) because you would not only need less land for the same amount of solar panels but also less solar panels for the same amount of energy. The ground for one tilted solar panel covering the same amount of sunrays as one laying flat should be the area of that solar panel (less by a factor of cos(40)) times the cos(40) (because its tilted) so its cos(40)^2 ≈ 59% so 41% of land area would be saved

Benedikt Jeschke - 2 years, 6 months ago

This is a goofy question. I assumed you could connect panel to panel and who cares where the shadow falls. You can definitely save space. The question should be worded that each panel must be 'on the ground' or 'have at least one edge in contact with the ground'.

Kurt Schwind - 2 years, 6 months ago

Yes, the answer is 1-cos(40) or 23%, unless there is a dispute with neighbours over the "right to light". The questioner should've been clear that shadows must remain within land you own. Question is also simplistic by just considering the effect at midday anyway. Not up to Brilliant's usual unambiguous standard.

Andy Boston - 2 years, 6 months ago

+1 for agreement. I assumed 1 large solar panel.

Shawn Wheatley - 2 years, 6 months ago

Yes, you can use a large solar panel, and you can save that 23%, there is a lot of ambiguity in this question

Diego Salcedo Tolosa - 2 years, 6 months ago

I agree with Richard, the problem does state that we must use commercially available panels and that the farm must form a square, but the panels could be raised to maximize the land use. The other piece of reality missing from the question is the earths 23 degree inclination to our orbit. To maintain maximum efficiency the angle of inclination on the panels needs to be reduced up to 23 degrees in the summer, further reducing land savings to (1 - cos(17))

Larry Allan - 2 years, 6 months ago

The problem description probably has been altered. By now, the problem is clear that we are using commercial panels (i.e. not arbitrarily large) and placing all of them on the ground with one edge, i.e. if you place them too closely together they will be in each other's shadow. In all cases, most of the shadow cannot be made to fall on the neighbor's land.

Roland van Vliembergen - 2 years, 6 months ago

It was a slightly badly posed question. I also assumed you could have one giant solar panel. In fact, if you do that then the answer is actually 1-cos^2(40). You get a square because a) the angle means less flat space but b) your panel is more efficient so you need less area facing the sun (think about energy per unit area in the flat vs angled case).

Anyway, the play devils advocate, I guess it's dumb to assume I can build a giant panel. But that's the difference between pure and applied maths I guess :)

Andrew Parker - 2 years, 6 months ago

I used exactly the same line of thinking as Richard did

Brett Fricke - 2 years, 6 months ago

Laszlo Mihaly
Nov 2, 2018

It does not matter how we tilt the panels, we cannot capture more power than the amount reaching the ground before we started to build the solar farm. On the other hand, tilting saves in terms of the total area of the solar panels we need. Since panels are expensive, most solar farms have tilted panels.

It is interesting that we can collect all the power only if we have a complete, flat coverage of the ground. No matter how we tilt the panels we will loose a fraction of the power, because the sunlight is coming from different directions during the day.

But what if the solar panels where different heights?

Itamar Ti - 2 years, 7 months ago

Does the light from the sun not reflect in different amounts depending on the angle of incidence?

Zac Mann - 2 years, 7 months ago

I would assume the solar panels would be covered with an anti-reflection coating, so that the amount of reflected light in all cases is only a very small fraction of the light. But without such measures indeed properly oriented panels would reflect much less light than flat-lying panels.

Roland van Vliembergen - 2 years, 6 months ago

Interesting, thank you.

Zac Mann - 2 years, 6 months ago

In that case, building solar farms on the north pole would be just as efficient as on the equator...which is astronomical nonsense. The correct solution is geometrical and doesn't assume that the Sun can cast light at all angles for an equal amount of time per angle.

lovro cupic - 2 years, 7 months ago

Aside from the obvious downside that solar panels also warm up in the sun, and would thus heat up their surroundings (melting the ice cap on the pole), as long as you do not consider the influence of our atmosphere this geometrical solution indeed is correct in saying that solar panels everywhere on earth could be equally efficient, as long as you angle them properly. Actually, due the working principles of these panels, they work more efficiently when they are cooler, so probably near the poles they'd even be more efficient.

However, the problem does not care about the number of solar panels being used, which means that laying them flat (and using almost twice as many) ultimately covers the same area with the solar farm. In practice you would never buy twice the number of panels if you could save half by putting them at the right angle. And of course near the poles you should almost place them vertically if you want to use the least number of panels for the same output.

The reason why panels near the equator in the end are more efficient, is the absorption and scattering of light in our atmosphere. Compare looking into the sun at sunset (relatively safe) to doing the same in the middle of the day with the sun high in the sky (don't try this without serious eye protection, such as eclipse glasses!) The smaller the path the light travels through our atmosphere the more intense it is. I.e. near the equator solar panels will be more efficient, as through the year the sunlight on average needs to travel the least distance through our atmosphere.

Roland van Vliembergen - 2 years, 6 months ago

Michael Mendrin
Nov 1, 2018

Let's look at this from the Sun's point of view. How would it even know if the solar panels are up at 40° or flat on the ground? Looks the same, it can't see any of the ground.

I do believe there is a flaw in this question: it doesn't really state the shape you solar farm should have. If you put all your solar pannels on one row, you would indeed save some land area.

Hugo ANTOINE - 2 years, 7 months ago

Really great point Hugo! I've edited the question to specify the shape of the solar farm, hopefully this addresses your concerns. If not, let me know and I'll iterate further.

Thanks for your clever lateral thinking. 🙏

Blake Farrow Staff - 2 years, 7 months ago

You could still have one giant square panel, which would save 1-cos(40) area (but there is no corresponding answer for this)

C Huang - 2 years, 7 months ago

Andrew Normand
Nov 12, 2018

At the risk of pedantry, the maximum sunlight intensity will be available at all latitudes between the tropics of cancer and capricorn, depending on the time of year.

Денис Лочмелис
Nov 12, 2018

This even makes things worse since each panel will have a shadow covering more space than a flat panel.

On the other hand, tilted panels collect sunlight more efficiently, and therefore they are used exactly like that.

Jaesung Lee
Nov 14, 2018

If tilt panel, shadow is existed by sun, so you have to consider the shadow. It is 1.30 times more longer than the length of panel at noon. Therefore there is no saved area.

Paul Cockburn
Nov 12, 2018

There is only a certain amount of sunlight falling on the land, so whether the panels are flat or tilted, they can only capture that much sunlight...

EXCEPT for the line of panels at the 'far end' of the solar farm. By tilting the last row such that the top of the panels are directly above the land boundary, these panel can capture an extra bit of sunlight that would, if the panels were flat, fall outside the land boundary. So the answer is a small amount of land area can be saved .

This small advantage can be increased by using bigger panels.

(Also note that tilting the panels may not save any significant land area, but it will at least reduce the number of panels needed.)

Apart from Andrew Normand only noon has been considered. Motorised panels would be needed for full efficiency, to face the panels from East to West throughout the day. Effectively reducing post available from the panels. Could get complex!

Philip Jackson - 2 years, 7 months ago

Sorry,power not post.

Philip Jackson - 2 years, 7 months ago

I made the silly assumption that the solar panels are all connected and for getting higher and higher from the ground, meaning it would use the sun that would have hit the ground outside of its ground area, which means it would get more energy.

Charles West - 2 years, 7 months ago

Solar farms don't belong in Norway

8 solutions

0 pending reports