Magnetic Field Surrounding a Straight Conductor

AI Thread Summary
The discussion focuses on understanding the derivation of the sine function in the context of the magnetic field surrounding a straight conductor. Participants clarify the relationship between the angles in the cross product ##\vec{ds} \times \hat r## and the angle ##\theta## depicted in a diagram. The connection is established through the definition of the cross product, leading to the equation involving sine functions. The clarification helps participants grasp the geometric relationships involved in the problem. Overall, the thread emphasizes the importance of understanding angles in vector calculations related to magnetic fields.
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1673658720024.png

Part of the solution is,
1673658743775.png

1673658803501.png

However, would someone please tell me where they got the sine function circled in red from?

Many thanks!
 

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Callumnc1 said:
However, would someone please tell me where they got the sine function circled in red from?
You're working with the cross product ##\vec{ds} \times \hat r##. How is the angle between ##\vec{ds}## and ##\hat r## related to the angle ##\theta## shown in diagram ##a##?
 
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TSny said:
You're working with the cross product ##\vec{ds} \times \hat r##. How is the angle between ##\vec{ds}## and ##\hat r## related to the angle ##\theta## shown in diagram ##a##?
Thank you @TSny ! I see how they got theta now. :) If we call the angle between ds and r hat as theta 2, then from the definition of the cross product:

dxsin(theta 2) = dxsin(pi/2 - theta) where I found theta 2 using angles in a triangle add up to 180 degrees.
 
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