B-field for a half-infinitely long wire

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    B-field Wire
In summary, the solution to this problem is to use the Biot-Savart Law. However, you need to consider the r^2 term when calculating the limits of integration, as the wire is infinitely long.
  • #1
ChiralSuperfields
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1674757975575.png

The solution is
1674757999684.png


However, I would like to understand how to solve this using Biot–Savart Law.

So far my working is:

## \vec {dB} = \frac {\mu_0Ids\sin\theta} {4\pi r^2}##

However, I'm not sure what to do about the ## r^2 ## since the wire is infinitely long. I am thinking about having the limits of integration to be ## \theta_1 = 0 ## and ## \theta_2 = \frac {\pi} {2} ##

Many thanks!
 
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  • #2
Callumnc1 said:
.So far my working is:

## \vec {dB} = \frac {\mu_0Ids\sin\theta} {4\pi r^2}##
Knowing a formula is of no value unless you know what all the variables mean and in what context the formula applies.
Please state what those variables mean here. From that you should be able to answer your question.
 
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  • #3
As your LHS is vector, RHS should be vector also. Check it out.
 
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  • #4
haruspex said:
Knowing a formula is of no value unless you know what all the variables mean and in what context the formula applies.
Please state what those variables mean here. From that you should be able to answer your question.
Thank you for your reply @haruspex!

##\mu_0## is the magnetic permeability of free space
##I## is the current in the vertical wire
## ds ## is the length element of the wire
## r^2 ## is the distance from each ##ds## to point ##P##

I think the thing that was tripping me up was that I thinking that we could call the infinite length to be ##a## which would mean that ##\tan\theta = \frac {x} {a} ## however this would be zero since ## a \rightarrow \infty ## then ##\tan \theta \rightarrow 0 ##. I was then going to solve for ##\theta## and have limits of integration to be angles ## \theta_1 = 0 ## and ## \theta_2 = \frac {\pi} {2} ##. However now I think I should get ## r ## in terms of ## \sin\theta ##. I will try to solve that now. Is my reasoning correct so far?
 
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  • #5
anuttarasammyak said:
As your LHS is vector, RHS should be vector also. Check it out.
Thanks @anuttarasammyak ! Sorry both sides should have ##-\hat k## as the direction.
 
  • #6
Callumnc1 said:
I should get ## r ## in terms of ## \sin\theta ##.
Yes.
 
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  • #7
1674778111259.png

1674778179068.png

1674778209637.png
 
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  • #8
haruspex said:
Yes.
Thank you @haruspex !
 
  • #10
Callumnc1 said:
Thank you for your solution @Alex Schaller! :)
You are welcome
 
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1. What is the B-field for a half-infinitely long wire?

The B-field, also known as the magnetic field, for a half-infinitely long wire is a measure of the strength and direction of the magnetic force around the wire. It is represented by the symbol B and is measured in units of Tesla (T).

2. How is the B-field calculated for a half-infinitely long wire?

The B-field for a half-infinitely long wire can be calculated using the formula B = μ₀I/2πr, where μ₀ is the permeability of free space, I is the current flowing through the wire, and r is the distance from the wire.

3. What is the direction of the B-field for a half-infinitely long wire?

The B-field for a half-infinitely long wire follows the right-hand rule, where if you point your thumb in the direction of the current flow, your fingers will curl in the direction of the B-field.

4. How does the B-field change as the distance from the wire increases?

The B-field decreases as the distance from the wire increases. This relationship follows an inverse square law, meaning that as the distance doubles, the B-field will decrease by a factor of four.

5. What are some real-world applications of the B-field for a half-infinitely long wire?

The B-field for a half-infinitely long wire has many practical applications, such as in electromagnets, electric motors, and generators. It is also used in medical imaging technology, such as MRI machines, to create detailed images of the body's internal structures.

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