How Does the Biot-Savart Law Apply to Angles in an Infinite Wire Calculation?

[cosmogrl]

In using the Biot-Savart Law for an finite wire, I am having trouble understanding the angles. I have sintheta1-sintheta2, where theta1 is measured from point P to the horizontal wire and from the vertical axis, to the left of point P. Theta 2 is measured to the right of point P.

I am ok until there. However, in going from that result to that for an infinite wire, my textbook (Serway and Jewett, Physics for scientists and engineers) says that as x goes to - infinity, theta 1 is (pi/2) and as x goes to positive infinity, theta 2 is (-pi/2). I seem to have those backwards, as theta1 being -pi/2 and theta2 being +pi/2. However, when I do it my way, I get an extra negative sign which should not be there. I am hoping someone can explain to my why theta is pi/2 to the left of point P and -pi/2 to the right of point P.

 

[Diptangshu]

In using the Biot-Savart Law for an finite wire, I am having trouble understanding the angles. I have sintheta1-sintheta2, where theta1 is measured from point P to the horizontal wire and from the vertical axis, to the left of point P. Theta 2 is measured to the right of point P.

I am ok until there. However, in going from that result to that for an infinite wire, my textbook (Serway and Jewett, Physics for scientists and engineers) says that as x goes to - infinity, theta 1 is (pi/2) and as x goes to positive infinity, theta 2 is (-pi/2). I seem to have those backwards, as theta1 being -pi/2 and theta2 being +pi/2. However, when I do it my way, I get an extra negative sign which should not be there. I am hoping someone can explain to my why theta is pi/2 to the left of point P and -pi/2 to the right of point P.

Here is the Picture of your Book.

if you want to check you are also Correct if you take θ₁, Start from the derivation at the Very beginning taking θ negative. You will get same Result.

 

[Kajal Sengupta]

I have a confusion. The way the diagram shows as in Diptangshu's answer theta increases in clockwise direction but conventionally anti clockwise direction is taken as positive. Can we change the convention as per our situation?

 

[Diptangshu]

I have a confusion. The way the diagram shows as in Diptangshu's answer theta increases in clockwise direction but conventionally anti clockwise direction is taken as positive. Can we change the convention as per our situation?

Here the main Issue is Direction of B. And B is Out of the Page as per r x ds direction...which is considered positive.

So you can use any measurement of θ as long as B is directed out of the page. Here the Reference is Vertical axis... You can take Right or Left as positive... And derive to get Same B.

 

[cosmogrl]

So, when using the equation as provided, is theta 1 always measured clockwise, but theta 2 is measured counterclockwise?

 

[Diptangshu]

So, when using the equation as provided, is theta 1 always measured clockwise, but theta 2 is measured counterclockwise?

Well if you take angle between r and ds, the limits will be π/2 to 0 and the Whole answer is multiplied by 2.
Which is the actual approach with no confusion... Because in Law of Biot Savart... dB = [μ₀i/4π] [i ds sinθ / r²]... The angle θ is angle between the Vectors r and ds.

So doesn't matter what is your direction... If you start from a direction and Approach any problem properly, you Get same results.

Better to take θ angle between r, ds to avoid Confusion.
The book has taken that for Symmetry.