How Can I Solve This Equation to Find the Correct Expression?

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In summary, In summary, the conversation involves solving for a variable in an equation and using approximations and algebraic manipulations to reach the correct expression.
  • #1
roam
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Homework Statement



I am trying to solve for ##P## in the equation:

$$Q=\frac{2RP}{\sqrt{\sigma_{T}^{2}+\left(2RPr\right)^{2}}+\sigma_{T}} \tag{1}$$

The correct answer must be:

$$\boxed{P=\frac{Q\sigma_{T}}{R(1-r^{2}Q^{2})}} \tag{2}$$

I am unable to get this expression.

Homework Equations

The Attempt at a Solution



Starting from (1), using the small ##x## approximation ##\sqrt{1+x^2} = 1+x^2/2## we can write it as:

$$\frac{2RP}{\sigma_{T}\sqrt{1+\left(\frac{2RPr}{\sigma_{T}}\right)}+\sigma_{T}}=\frac{2RP}{2\sigma_{T}+\frac{2R^{2}P^{2}r^{2}}{\sigma_{T}}}$$

$$\left(\frac{2QR^{2}r^{2}}{\sigma_{T}}\right)P^{2}-(2R)P+2Q\sigma_{T}=0$$

Using the quadratic formula gives:

$$P=\frac{(1-Q^{2}r^{2})\sigma_{T}}{QRr^{2}}$$

Clearly, this doesn't agree with (2). Also, I was told that I shouldn't use the small value approximation and that I should not get a quadratic. But without using the approximation, I still get a quadratic. :confused:

So how can I get to the correct expression?

Any explanation would be greatly appreciated.

P. S. This equation relates to optical receivers where ##P## is the sensitivity and ##\sigma_T## and ##2RPr## are the thermal and intensity noise respectively.
 
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  • #2
Multiply both sides by the denominator of the RHS. Subtract QσT from both sides. Square both sides. then see how you get on.
The key to this is isolating the square root expression on one side of the equation, if possible, then squaring everything.
 
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  • #3
Try starting out by multiplying numerator and denominator of the RHS by ##\sqrt{\sigma_{T}^{2}+\left(2RPr\right)^{2}}-\sigma_{T}##. Then isolate ##\sqrt{\sigma_{T}^{2}+\left(2RPr\right)^{2}}-\sigma_{T}## and ##\sqrt{\sigma_{T}^{2}+\left(2RPr\right)^{2}}+\sigma_{T}## on the left hand sides of two equations, and add the two equations together.
 
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  • #4
Thank you very much. Here is what I got so far:

$$Q\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}=2RP-Q\sigma_{T}$$

$$Q^{2}\sigma_{T}^{2}+(2QrRP)^{2}=\left(2RP\right)^{2}-Q^{2}\sigma_{T}^{2}$$

$$4R^{2}(Q^{2}r^{2}-1)P^{2}=-2\left(Q\sigma_{T}\right)^{2}$$

$$P=\frac{Q\sigma_{T}}{R\sqrt{2\left(1-Q^{2}r^{2}\right)}}$$

It's now very close to the required expression. How can I get rid of the square root and the factor of 2 in the denominator?
 
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  • #5
Chestermiller said:
Try starting out by multiplying numerator and denominator of the RHS by ##\sqrt{\sigma_{T}^{2}+\left(2RPr\right)^{2}}-\sigma_{T}##. Then isolate ##\sqrt{\sigma_{T}^{2}+\left(2RPr\right)^{2}}-\sigma_{T}## and ##\sqrt{\sigma_{T}^{2}+\left(2RPr\right)^{2}}+\sigma_{T}## on the left hand sides of two equations, and add the two equations together.

I have tried this approach:

$$Q=\frac{2RP}{\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}+\sigma_{T}}\frac{\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}-\sigma_{T}}{\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}-\sigma_{T}}$$

$$Q=\frac{2RP\left(\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}-\sigma_{T}\right)}{\sigma_{T}^{2}+(2RPr)^{2}-\sigma_{T}^{2}}$$

$$\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}-\sigma_{T}=2QRr^{2}P \tag{i}$$

$$\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}+\sigma_{T}=\frac{2RP}{Q} \tag{ii}$$

Adding the two equations:

$$2\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}=\frac{2RP}{Q}+2QRr^{2}P$$

$$\left(2RrP\right)^{2}=\frac{R^{2}(1+Q^{2}r^{2})^{2}P^{2}}{Q^{2}}-\sigma_{T}^{2}$$

I wasn't able to proceed further from here to get to equation (2). Any suggestions? :confused:
 
  • #6
roam said:
Thank you very much. Here is what I got so far:

$$Q\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}=2RP-Q\sigma_{T}$$

$$Q^{2}\sigma_{T}^{2}+(2QrRP)^{2}=\left(2RP\right)^{2}-Q^{2}\sigma_{T}^{2}$$

$$4R^{2}(Q^{2}r^{2}-1)P^{2}=-2\left(Q\sigma_{T}\right)^{2}$$

$$P=\frac{Q\sigma_{T}}{R\sqrt{2\left(1-Q^{2}r^{2}\right)}}$$

It's now very close to the required expression. How can I get rid of the square root and the factor of 2 in the denominator?
You seem to have lost a term in going from the first equation above to the second.
 
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  • #7
haruspex said:
You seem to have lost a term in going from the first equation above to the second.

Thank you very much. It solved the problem.
 
  • #8
From Post #5:
roam said:
$$\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}-\sigma_{T}=2QRr^{2}P \tag{i}$$
$$\sqrt{\sigma_{T}^{2}+(2RPr)^{2}}+\sigma_{T}=\frac{2RP}{Q} \tag{ii}$$
Rather than adding these equations, subtract Eq. (i) from Eq. (ii) .
 
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Related to How Can I Solve This Equation to Find the Correct Expression?

What is an equation?

An equation is a mathematical statement that shows the equality between two expressions, typically involving one or more variables.

What is the purpose of solving an equation?

The purpose of solving an equation is to find the value(s) of the variable(s) that make the equation true. This allows us to solve problems and make predictions in various fields, such as science, engineering, and economics.

What are the basic steps for solving an equation?

The basic steps for solving an equation are: 1) Simplify both sides of the equation by combining like terms, 2) Isolate the variable by performing inverse operations, 3) Check the solution by substituting it back into the original equation, and 4) If the solution is not valid, continue solving by repeating the previous steps.

What are inverse operations?

Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, as well as multiplication and division. In solving an equation, we use inverse operations to isolate the variable and find its value.

How do I know if my solution is correct?

You can check your solution by substituting it back into the original equation and simplifying both sides. If the resulting values on both sides are equal, then your solution is correct. If they are not equal, you may need to continue solving or check for any possible errors in your calculations.

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