Dielectric constant of a rod

In summary, the dielectric constant should be such that even when the incident angle is slightly less than \frac { \pi} 2, the ray should come out of the rod without getting absorbed. This matches what @Pushoam has done.
  • #1
Pushoam
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Homework Statement


upload_2017-12-28_12-27-28.png


Homework Equations

The Attempt at a Solution


## n_1 = \sqrt{ \epsilon _1} ## ...(1) ,

## n_2 = 1 ## ...(2)

## \frac { \sin {\theta_i}}{ \sin {\theta_l} } = \frac { n_1}{n_2} = n_1 ## ...(3)

## \cos{\theta_1} = \frac { n_2}{n_1} = \frac1{ n_1} ## ...(4)

According to the question, the dielectric constant should be such that even when the incident angle is slightly less than ## \frac { \pi} 2 ## , the ray should come out of the rod without getting absorbed.

So, ## \theta _i \leq \frac { \pi} 2 ## ...(5)

Taking ##\theta _i = \frac { \pi} 2 ## , ...(6) I don't know why I am taking this. ## \frac1{ \sin {\theta_l} }= n_1 ## ...(7)

## \cos{\theta_1} = \frac1{ n_1} ## ...(8)

From (7) and (8), I get

## n_1 = \sqrt{ 2} ## ...(9)

From (1), ## \epsilon _1 = n_1^2 ## ...(10)

So, that dielectric constant = 2, option (c).

Is this correct?
 

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  • #2
Snell's law is written ## n_1 \sin(\theta_1)=n_2 \sin(\theta_2) ##. For this case ## n_1=\sqrt{\epsilon_{r1}} ##, and ## n_2 =1 ##. ## \\ ## ## \theta_1 ## is the angle of incidence inside the material, measured from the normal. There is a ## \theta_1 ## for which the left side of the equation is equal to 1. What happens if ## \theta_1 ## is such that the left side of the equation is greater than 1 ? Can you get a solution for the right side in that case, to determine the emerging angle of the refracted ray? ## \\ ## Editing: This problem is slightly tricky: If ## \theta_i=90^o ##,(the steepest angle of incidence at the entry point), ## \theta_r=\sin^{-1}(1/n_1) ##. The resulting ## \theta_1=90^o-\theta_r ##. I'll let you try to finish it up. Meanwhile, the file you uploaded is apparently the wrong one.
 
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  • #3
Charles Link said:
Snell's law is written ## n_1 \sin(\theta_1)=n_2 \sin(\theta_2) ##. For this case ## n_1=\sqrt{\epsilon_{r1}} ##, and ## n_2 =1 ##. ## \\ ## ## \theta_1 ## is the angle of incidence inside the material, measured from the normal. There is a ## \theta_1 ## for which the left side of the equation is equal to 1. What happens if ## \theta_1 ## is such that the left side of the equation is greater than 1 ? Can you get a solution for the right side in that case, to determine the emerging angle of the refracted ray? ## \\ ## Editing: This problem is slightly tricky: If ## \theta_i=90^o ##,(the steepest angle of incidence at the entry point), ## \theta_r=\sin^{-1}(1/n_1) ##. The resulting ## \theta_1=90^o-\theta_r ##. I'll let you try to finish it up. Meanwhile, the file you uploaded is apparently the wrong one.
As far as I can make out, this matches what @Pushoam has done. Have you identified an error in the working?
 
  • #4
haruspex said:
As far as I can make out, this matches what @Pushoam has done. Have you identified an error in the working?
The OP's statement after equation (6) was somewhat confusing. Meanwhile, I was expecting to see the statement ## n_1 \sin(\theta_1)>1 ## for total internal reflection. In addition, ## \sin(\theta_1)=\cos(\theta_r)=\frac{\sqrt{n_1^2-1}}{n_1} ##. Thereby ## \sqrt{n_1^2-1}>1 ##, so that ## n_1^2=\epsilon_{r1}>2 ##. ## \\ ## I didn't see these details in the OP's solution. ## \\ ## (It was difficult to answer the question by @haruspex without providing the solution.) ## \\ ## From what I can see, the OP gets the right answer, but the algebraic steps to the answer are not readily apparent. Perhaps the OP performed a similar algebra, but too many steps were omitted to readily tell how the OP arrived at the answer. :) ## \\ ## In any case, I think the OP @Pushoam might find these details useful.
 
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  • #5
Charles Link said:
The OP's statement after equation (6) was somewhat confusing. Meanwhile, I was expecting to see the statement ## n_1 \sin(\theta_1)>1 ## for total internal reflection. In addition, ## \sin(\theta_1)=\cos(\theta_r)=\frac{\sqrt{n_1^2-1}}{n_1} ##. Thereby ## \sqrt{n_1^2-1}>1 ##, so that ## n_1^2=\epsilon_{r1}>2 ##. ## \\ ## I didn't see these details in the OP's solution. ## \\ ## (It was difficult to answer the question by @haruspex without providing the solution.) ## \\ ## From what I can see, the OP gets the right answer, but the algebraic steps to the answer are not readily apparent. Perhaps the OP performed a similar algebra, but too many steps were omitted to readily tell how the OP arrived at the answer. :) ## \\ ## In any case, I think the OP @Pushoam might find these details useful.
Ok, thanks for clarifying.
 
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1. What is the dielectric constant of a rod?

The dielectric constant of a rod is a measure of its ability to store and transmit electric charge. It is also known as the relative permittivity, and is typically denoted by the symbol εr. It describes how easily an electric field can pass through a material compared to a vacuum.

2. How is the dielectric constant of a rod calculated?

The dielectric constant of a rod can be calculated by dividing the capacitance of the rod by the capacitance of a vacuum capacitor with the same dimensions. It can also be calculated by dividing the electric displacement field in the rod by the electric field in a vacuum.

3. What factors can affect the dielectric constant of a rod?

The dielectric constant of a rod can be affected by factors such as the type of material the rod is made of, its dimensions, and the presence of any impurities or defects in the material. Temperature and frequency of the electric field can also have an impact on the dielectric constant.

4. How does the dielectric constant of a rod affect its electrical properties?

The dielectric constant of a rod plays a crucial role in determining its electrical properties, such as its ability to store and transmit electric charge, its capacitance, and its resistance to electric fields. A higher dielectric constant means the rod can store more charge and is more resistant to electric fields.

5. What are some common applications of the dielectric constant of a rod?

The dielectric constant of a rod is important in various fields such as electronics, telecommunications, and material science. It is used in the design and analysis of capacitors, transmission lines, and other electronic components. It is also a key parameter in the study of insulation materials and their effectiveness in preventing electric current flow.

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