How does water rise along a glass plate? (surface tension question)

• Amitayas Banerjee
In summary: This is the exact pressure needed to support the meniscus.In summary, the conversation discusses the phenomenon of water rising along a glass plate and the attempt to find a mathematical interpretation for it. The individual looked for solutions in books and online, and eventually found a contradiction in their own mathematical analysis. However, the expert summarizes that the analysis was correct, but the sign of the pressure force was incorrect. The correct analysis shows that the pressure within the meniscus region is sub-atmospheric, and this is the exact pressure needed to support the meniscus.
Amitayas Banerjee
So, I was studying about general properties of matter and topics like surface tension. I came across the phenomenon of water rising along a glass plate like in the picture. I looked for some mathematical interpretation of this on the internet and in some books.

[![enter image description here][1]][1]

I looked for some mathematical interpretation of this on the internet and in some books.
I found some mathematical understanding of the phenomenon in the book **Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves** and also elaborate answers on StackExchange like this one: https://physics.stackexchange.com/q...an-water-rise-above-the-edge-of-a-glass/45122

But I decided to find the height along which the water climbs on the glass by balancing forces on the **infinitely long** water element:

[![enter image description here][2]][2]

It is to be noted that the height of this water element is **$h$** and it has an infinite length in the horizontal direction.

Now the pressure force $P$ can be calculated as
$$P=\int_0^h \rho gz dz=\frac{1}{2}\rho g h^2$$

On balncing forces in the horizontal direction, we get $$P+S =S\sin \theta$$ $$\Rightarrow \frac{1}{2}\rho g h^2= S(\sin \theta -1)$$ which is surely a **contradiction** as the term in the left hand side is bound to be positive. Hence I believe that I have apparently disproved the fact that water would rise along the glass plate. But I also know that this is true that water has to rise as evident from daily experiences. So, where does my math go wrong?

[1]: https://i.stack.imgur.com/FjxSP.png
[2]: https://i.stack.imgur.com/e6BNY.png

Your analysis is fine, except for the sign of P which should be negative rather than positive, because the pressure within the meniscus region is sub-atmospheric. The pressure increases with depth within the meniscus (from a sub-atmospheric value at the top), until it reaches atmospheric at the bottom.

If you would like me to present a detailed analysis of this straightforward problem, I can. But, again, there is nothing wrong with your assessment, aside from the sign of P.

Again, it is as simple as this: If z is the depth measured downward from the top of the meniscus, then z = h represents the horizontal surface where the gauge pressure is zero. So, within the meniscus, the gauge pressure at depth z is ##-\rho g (h-z)##, so that at z = 0, the gauge pressure is ##-\rho g h##.

vanhees71 and Orodruin

1. How does surface tension cause water to rise along a glass plate?

Surface tension is a phenomenon that occurs when the molecules of a liquid are more attracted to each other than to the molecules of the surrounding surface. In the case of water, the molecules are strongly attracted to each other due to hydrogen bonding. This creates a cohesive force that pulls the water molecules together, causing them to form a thin layer on the surface of the glass plate. This layer then pulls on the water molecules below it, causing the water to rise along the plate.

2. Why does water rise higher along the edges of the glass plate?

Water molecules are more strongly attracted to the surface of the glass than to each other. This creates an adhesive force that pulls the water molecules towards the glass surface. As the water rises along the plate, the adhesive force becomes stronger near the edges, causing the water to rise higher in those areas.

3. Does the temperature of the water affect its ability to rise along the glass plate?

Yes, the temperature of the water does have an impact on its ability to rise along the glass plate. As the temperature of the water increases, the molecules move faster and have more energy. This makes it easier for the water molecules to overcome the cohesive forces and break the surface tension, resulting in less water rising along the plate.

4. Can other liquids besides water rise along a glass plate due to surface tension?

Yes, other liquids can also rise along a glass plate due to surface tension. However, the strength of the surface tension and adhesive forces may vary depending on the type of liquid. For example, liquids with weaker cohesive forces may not rise as high as water along a glass plate.

5. Is there a limit to how high water can rise along a glass plate due to surface tension?

Yes, there is a limit to how high water can rise along a glass plate due to surface tension. This limit is known as the capillary rise height and is determined by the balance between the cohesive and adhesive forces of the water molecules. The narrower the glass plate, the higher the water can rise due to the increased surface tension. However, once the capillary rise height is reached, the weight of the water will overcome the surface tension, causing it to overflow.

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