Decomposition of linearly polarized field MRI

[Jen2114]

Homework Statement

Hi, I am having trouble understanding how the B1 field as described by (3.48) in the image attached in MRI which is described as a linearly polarized field is decomposed into it's final two circularly polarized field as described by (3.49) in the image attached.

Homework Equations

The Attempt at a Solution

I understand that because it's 2B1 that you have one in the clockwise direction and one in the counterclockwise direction. That's why the first B1 has a negative in front of the sine term and the second doesn't. However, how do you get a cosine and sine term if in the first equation (3.48) if you only have a cosine to begin with. I know you describe a circle like the unit circle with cosine and sine but I still don't understand why a term with only cosine is expanded to cosine and sine. Clarification on how to go from 3.48 to 3.49 would be greatly appreciated.

 

[DrClaude]

However, how do you get a cosine and sine term if in the first equation (3.48) if you only have a cosine to begin with.

You simply add and subtract the same term:
$$
2a = 2a + b - b = (a+b) + (a-b)
$$
This is a common trick used when you want to introduce an additional term, just like multiplying by $b/b$.

 

[Jen2114]

You simply add and subtract the same term:
$$
2a = 2a + b - b = (a+b) + (a-b)
$$
This is a common trick used when you want to introduce an additional term, just like multiplying by $b/b$.

Thank you ! I wasn't aware of this but now that I am it makes much more sense!