Magnetic Field Surrounding a Straight Conductor

member 731016 · Jan 13, 2023

For this problem,

Textbook heading reading "Example 29.1 Magnetic Field Surrounding a Thin, Straight Conductor" followed by a problem statement about a thin straight wire of finite length carrying constant current I along the x axis, asking for the field at point P.

Part of the solution is,

Textbook solution excerpt headed SOLUTION, reading Conceptualize: from the Biot-Savart law the field magnitude is proportional to the current and decreases with distance a from the wire to point P, depending on angles θ1 and θ2 in Figure 29.4b, with origin at O and P along the positive y axis.

Textbook figure of a thin straight wire carrying current I with diagrams of the angles theta and theta-2 from point P to the wire ends, plus a derivation step with a circled sine term.

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

Many thanks!

TSny · Jan 13, 2023

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##?

member 731016 · Jan 13, 2023

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.

Magnetic Field Surrounding a Straight Conductor

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