Problem on Radiation Pressure

[Mohammad Ishmas]

Homework Statement

There is a Conical structure such that its base is completely reflecting and the tip of the cone which has a very small area and completely absorbent surface, if the light of same intensity is incident on the tip and on the base simultaneously then find the ratio of the force exerted by light at the base to the force exerted by it on the tip

Relevant Equations

F = (2I*A/c) for completely reflecting surfaces and F = I*A/C for absorbent surfaces , where I is the intensity and c is the speed of light and A is the area on which the light is being incident.

Question on radiation pressure

 

[WWGD]

Homework Statement: There is a Conical structure such that its base is completely reflecting and the tip of the cone which has a very small area and completely absorbent surface, if the light of same intensity is incident on the tip and on the base simultaneously then find the ratio of the force exerted by light at the base to the force exerted by it on the tip
Relevant Equations: F = (2I*A/c) for completely reflecting surfaces and F = I*A/C for absorbent surfaces , where I is the intensity and c is the speed of light and A is the area on which the light is being incident.

Question on radiation pressure

What have you tried so far? Please show your work.

 

[Mohammad Ishmas]

I have figured out that the ratio will depend only on areas of the base and the tip since the intensity of the light is same, however I am clueless in finding the area of the base in terms of area of the tip or vice versa

 

[Steve4Physics]

I have figured out that the ratio will depend only on areas of the base and the tip since the intensity of the light is same, however I am clueless in finding the area of the base in terms of area of the tip or vice versa

The tip of a cone is a point, so its area is zero. Do you mean a truncated cone?

The question seems incomplete. You need to post the exact, complete original question including any diagram(s).

 

[Mohammad Ishmas]

The tip of a cone is a point, so its area is zero. Do you mean a truncated cone?

The question seems incomplete. You need to post the exact, complete original question including any diagram(s).

this is the original question itself, and by "very small area" it means a area which is infinitesimally small, also there are no diagram(s)

 

[Steve4Physics]

this is the original question itself, and by "very small area" it means a area which is infinitesimally small, also there are no diagram(s)

So what do you think the force (due to radiation pressure) on an infinitesimally small area will be?

 

[Mohammad Ishmas]

So what do you think the force (due to radiation pressure) on an infinitesimally small area will be?

Let us consider the infinitesimally small area to be 'dA' , then we can directly use the formula I have mentioned that is F = (I*A)/c (This is for completely absorbent surfaces, where I is intensity, A is area which in this case is 'dA' and c is the speed of light )

 

[Steve4Physics]

Let us consider the infinitesimally small area to be 'dA' , then we can directly use the formula I have mentioned that is F = (I*A)/c (This is for completely absorbent surfaces, where I is intensity, A is area which in this case is 'dA' and c is the speed of light )

If the total area is infinitesimally small, then there is no integration to perform; the total force is infinitesimally small, i.e zero. A non-zero force can only arise for a finite area.

Are trying to solve a different problem requiring integration? Maybe this:

 

[Mohammad Ishmas]

If the total area is infinitesimally small, then there is no integration to perform; the total force is infinitesimally small, i.e zero. A non-zero force can only arise for a finite area.

Are trying to solve a different problem requiring integration? Maybe this:

View attachment 359228

Yeah you are right the diagram would be like this , and it would require integration but how can I find the area of the base in terms of the area of the tip (basically the question has a pre made assumption that the tip has a area that is very small)

 

[jbriggs444]

how can I find the area of the base in terms of the area of the tip

You cannot.

Not without additional information. Which you claim does not exist.

 

[Mohammad Ishmas]

You cannot.

Not without additional information. Which you claim does not exist.

Can't we use integration ?? , the area of the tip will coincide to some area of the base. And what additional information would be required to find the area of base in terms of area of tip

 

[Steve4Physics]

Can't we use integration ?? , the area of the tip will coincide to some area of the base. And what additional information would be required to find the area of base in terms of area of tip

The 'area of the tip' is zero. Do you mean the 'area of the curved side' (shown in black in the Post #8 diagram)?

 

[Mohammad Ishmas]

The 'area of the tip' is zero. Do you mean the 'area of the curved side' (shown in black in the Post #8 diagram)?

I am sorry, the question actually says a that there is a Frustrum of a cone with the area of the top part (that I called tip earlier) being very small . I had assumed that area to be 'dA' , However how can I find area of the base in terms of dA ??

 

[Steve4Physics]

I am sorry, the question actually says a that there is a Frustrum of a cone with the area of the top part (that I called tip earlier) being very small . I had assumed that area to be 'dA' , However how can I find area of the base in terms of dA ??

A diagram would have saved a lot of confusion! The area of the small face is small but finite - so you shouldn't represent it using 'dA'. That's misleading.

There is still too much missing information. What values (lengths and maybe angles) do you think are needed to answer the question?

 

[Mohammad Ishmas]

A diagram would have saved a lot of confusion! The area of the small face is small but finite - so you shouldn't represent it using 'dA'. That's misleading.

There is still too much missing information. What values (lengths and maybe angles) do you think are needed to answer the question?

I think I will need radius of both the base and the frustum (the top part), and the heights also. I am thinking of a approach to find the rate at which the area decreases as we move from bottom to top of the frustum of the ocne

 

[jbriggs444]

I think I will need radius of both the base and the frustum (the top part), and the heights also.

If you have the radius of the base, you can calculate the area of the base, right?

If you have the radius of the frustrum, you can calculate the area of the frustrum, right?

Why would you need the height? We are not told how radiation affects the sides of the truncated cone. So knowing the area of the sides tells us nothing useful.

 

[Mohammad Ishmas]

If you have the radius of the base, you can calculate the area of the base, right?

If you have the radius of the frustrum, you can calculate the area of the frustrum, right?

Why would you need the height? We are not told how radiation affects the sides of the truncated cone. So knowing the area of the sides tells us nothing useful

In the original question, radii and heights aren't given. We basically have to just derive a ratio of the forces , and the best approach I could think of was to just find the area of the base in terms of area of the frustrum. It would give us something in terms of either radii or heights , but how exactly do we do that ?

 

[Steve4Physics]

One approach when given a very badly formulated question (which, IMO, this is) is to present your own best-guess interpretation of the question and then answer that. Here’s my best-guess interpretation of the question:

A right conical frustum has two circular flat faces, A and B.
Face A is the larger, with radius $R_A$ and is a perfect reflector.
Face B, with radius $R_B$ is a perfect absorber.

As shown in the diagram, light of the same intensity, $I$, is normally incident on faces A and B.
.

Find the ratio of the magnitude of the force exerted by light on face A to the magnitude of the force exerted by light on face B.

Can you answer that?

EDIT: 'force' changed to 'magnitude of the force'.

 

[Mohammad Ishmas]

One approach when given a very badly formulated question (which, IMO, this is) is to present your own best-guess interpretation of the question and then answer that. Here’s my best-guess interpretation of the question:

A right conical frustum has two circular flat faces, A and B.
Face A is the larger, with radius $R_A$ and is a perfect reflector.
Face B, with radius $R_B$ is a perfect absorber.

As shown in the diagram, light of the same intensity, $I$, is normally incident on faces A and B.
.
View attachment 359247
Find the ratio of the magnitude of the force exerted by light on face A to the magnitude of the force exerted by light on face B.

Can you answer that?

EDIT: 'force' changed to 'magnitude of the force'.

Yes this is the question, I am sorry for not giving the diagram as it was not there in the original question

 

[haruspex]

One approach when given a very badly formulated question (which, IMO, this is) is to present your own best-guess interpretation of the question and then answer that. Here’s my best-guess interpretation of the question:

A right conical frustum has two circular flat faces, A and B.
Face A is the larger, with radius $R_A$ and is a perfect reflector.
Face B, with radius $R_B$ is a perfect absorber.

As shown in the diagram, light of the same intensity, $I$, is normally incident on faces A and B.
.
View attachment 359247
Find the ratio of the magnitude of the force exerted by light on face A to the magnitude of the force exerted by light on face B.

Can you answer that?

EDIT: 'force' changed to 'magnitude of the force'.

That is still a very strange question. In practice, how would it be that the light at the sharp end is only incident on the flat face? The version you suggested in post #8 is both more reasonable physically and more interesting algebraically.
@Mohammad Ishmas , please post the original question exactly as given to you, whatever language it is in.

 

[Steve4Physics]

Yes this is the question,

No it is not! In my version, the radii are given ($R_A$ and $R_B$) - you can use them in the answer. But in Post #17 you explicitly said:

In the original question, radii and heights aren't given

There are too many conflicting statements. Can you clarify things by doing what @haruspex asked in Post ##20?

post the original question exactly as given to you, whatever language it is in.

 

[Mohammad Ishmas]

No it is not! In my version, the radii are given ($R_A$ and $R_B$) - you can use them in the answer. But in Post #17 you explicitly said:


There are too many conflicting statements. Can you clarify things by doing what @haruspex asked in Post ##20?

The original question is as follows -

A uniform beam of light falls on a frustum of a cone. The smaller top surface of the frustum is completely absorbent, while the larger base is perfectly reflective. The original cone, before being cut, had a total height H and a base radius R. The frustum was formed by removing the upper part of the cone, which had a height h from the tip to the top of the frustum. Find the ratio of the force exerted by the light on the base to the force exerted on the top surface.


I am also sorry for not posting it earlier but I couldn't as this was in a regional language

 

[jbriggs444]

A uniform beam of light falls on a frustum of a cone. The smaller top surface of the frustum is completely absorbent, while the larger base is perfectly reflective. The original cone, before being cut, had a total height H and a base radius R. The frustum was formed by removing the upper part of the cone, which had a height h from the tip to the top of the frustum. Find the ratio of the force exerted by the light on the base to the force exerted on the top surface.

From this information one can calculate the radius of the frustrum, $r$.

Can you see how? Can you show us the formula you come up with?

 

[Mohammad Ishmas]

From this information one can calculate the radius of the frustrum, $r$.

Can you see how? Can you show us the formula you come up with?

Yeah I can see that , we can use similar triangles to find that r = R(h/H)

 

[Steve4Physics]

A uniform beam of light falls on a frustum of a cone. The smaller top surface of the frustum is completely absorbent, while the larger base is perfectly reflective.

It now sounds like there is only one beam of light. That makes previous descriptions wrong. Possibly the intended setup is this:

This would make much more sense.

 

[Mohammad Ishmas]

Yeah I can see that , we can use similar triangles to find that r = R(h/H)

I found a solution using this value of 'r', I just plugged this value of "r" in πr^2 , then you would get a ratio of the two areas in terms of height, basically if the area of the base in A and area of top surface is B , then ratio A/B = (H/h)^2 , then we can just use the formulas to find the ratio of the forces

 

[haruspex]

I found a solution using this value of 'r', I just plugged this value of "r" in πr^2 , then you would get a ratio of the two areas in terms of height, basically if the area of the base in A and area of top surface is B , then ratio A/B = (H/h)^2 , then we can just use the formulas to find the ratio of the forces

Did you take into account the difference in reflectivity?

 

[Mohammad Ishmas]

Did you take into account the difference in reflectivity?

Yeah I did , I used the different formulas for completely reflective and completely absorbent surfaces (for reflective surfaces the force just becomes twice). the final answer comes out to be , Fa/Fb = 2(H/h)^2

 

[haruspex]

Yeah I did , I used the different formulas for completely reflective and completely absorbent surfaces (for reflective surfaces the force just becomes twice). the final answer comes out to be , Fa/Fb = 2(H/h)^2

Ok. However, if the diagram in post #25 is accurate, the larger surface is partly shaded or displaced by the smaller one. But since the ratio is given as large, presumably we can ignore that.

 

[Mohammad Ishmas]

Ok. However, if the diagram in post #25 is accurate, the larger surface is partly shaded or displaced by the smaller one. But since the ratio is given as large, presumably we can ignore that.

Sorry but I didn't get you, what do you mean by , "the larger surface is partly shaded or displaced by the smaller one." ?

 

[jbriggs444]

Sorry but I didn't get you, what do you mean by , "the larger surface is partly shaded or displaced by the smaller one." ?

The portion of the light beam that strikes the smaller upper surface (the frustrum) is absorbed. It cannot then strike the larger lower surface. The lower surface is partly shaded by the upper surface.

Presumably the walls of the cone are constructed of a transparent material with negligible thickness or a refractive index of one.

It is important to assume that the base of the cone has specular reflection (like a mirror) rather than diffuse reflection (like a white painted wall). Not only does this assure us that the full factor of two is achieved, it also assures us that the light rebounding from the base does not strike the bottom side of the frustrum.

 

[Mohammad Ishmas]

Ok. However, if the diagram in post #25 is accurate, the larger surface is partly shaded or displaced by the smaller one. But since the ratio is given as large, presumably we can ignore that.

The portion of the light beam that strikes the smaller upper surface (the frustrum) is absorbed. It cannot then strike the larger lower surface. The lower surface is partly shaded by the upper surface.

Presumably the walls of the cone are constructed of a transparent material with negligible thickness or a refractive index of one.

It is important to assume that the base of the cone has specular reflection (like a mirror) rather than diffuse reflection (like a white painted wall). Not only does this assure us that the full factor of two is achieved, it also assures us that the light rebounding from the base does not strike the bottom side of the frustrum.

Thank you , I understood your point. It would have been more clearer if there were 2 light sources mentioned in the question like one that strikes the top of frustrum and the other strikes the base .

 

[Mohammad Ishmas]

If the total area is infinitesimally small, then there is no integration to perform; the total force is infinitesimally small, i.e zero. A non-zero force can only arise for a finite area.

Are trying to solve a different problem requiring integration? Maybe this:

View attachment 359228

If this was the question , where the light strikes the slanted surface of the cone then how we would find the area of where the light exerts force ?

 

[Steve4Physics]

If this was the question , where the light strikes the slanted surface of the cone then how we would find the area of where the light exerts force ?

@Mohammad Ishmas, there are now several different versions of your question on this thread. It is still not clear (to me anyway) which one (if any) is correct.

New replies to this thread are going to add to the confusion because it will not be clear which version of the question is being addressed.

If you have a specific question, then you could start a completely new thread with:
- an explanation that it is a ‘spin-off’ from this thread;
- a complete, accurate, unambiguous problem-statement, including a diagram;
- your own attempt at an answer - as required by the forum rules.