Magnetic field of infinite wire

In summary, the problem involves finding the magnitude of the magnetic field at point P, which is located a distance r from an infinite wire carrying current I. The equation used is dB = (μ/4π) (I dl×r )/r3, with the relevant distance r being given by Pythagoras' theorem. Further calculations involve determining the angle φ and making a substitution for y before integrating the expression for dB.
  • #1

Homework Statement


An infinite wire carries current I.

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I hope the picture works! The vertical arrow shows the direction of current in the wire. The green arrow has length a and ends at point P. What is the magnitude of the magnetic field at point P?

Homework Equations


dB = (μ/4π) (I dl×r )/r3
Where the r on the numerator is a position vector.

The Attempt at a Solution


The horizontal part of the wire is irrelevant, as l is in the same direction as r at that point.

r is the distance of P from the vertical wire, I think. I called the direction of the green arrow the positive x direction, the direction of current and of dl is the y direction. I have to find the magnitude of the field, so I have to integrate the expression for dB. dl can be replaced with dy as they are in the same direction.

I, μ and 4π are constants and can be taken out of an integral. The magnitude of the cross product on the numerator is given by r dy sinφ, where r now cancels with the r in the denominator.
I have to do (μI / 4π) ∫ (dy sinφ) / r2
So the distance r to the wire will be given by Pythagoras' theorem, so r = (x2+y2)1/2.
My major problem is, I don't really know which angle φ is. It's one of two! Is sinφ given by x/(x2+y2) or should y be on the numerator instead?

Anyway, I carried on assuming x is on the numerator. Then the integral becomes:
∫ x dy / (x2+y2)3/2

So do a substitution of y=xtanφ. And then how I carry on really depends on what angle φ is!
Thanks for any help :)
 
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  • #2
Oh, and all black lines indicate a part of the wire, which does turn a corner.
 

1. What is the concept of a magnetic field of infinite wire?

The magnetic field of an infinite wire refers to the magnetic field that is created by a current-carrying wire that extends infinitely in both directions. This magnetic field is perpendicular to the wire and follows a circular path around it.

2. How is the strength of the magnetic field of an infinite wire calculated?

The strength of the magnetic field of an infinite wire can be calculated using the formula B = (μ0 x I)/2πr, where μ0 is the permeability of free space, I is the current in the wire, and r is the distance from the wire.

3. Does the strength of the magnetic field decrease as you move away from the infinite wire?

Yes, the strength of the magnetic field decreases as you move further away from the infinite wire. This is because the magnetic field follows an inverse square law, meaning that the strength decreases as the distance from the source increases.

4. How is the direction of the magnetic field of an infinite wire determined?

The direction of the magnetic field of an infinite wire can be determined using the right-hand rule. If you point your thumb in the direction of the current flow in the wire, the direction of your fingers will indicate the direction of the magnetic field.

5. What are some real-life applications of the magnetic field of infinite wire?

The magnetic field of infinite wire has several real-life applications, such as in electromagnets, motors, and generators. It is also used in medical imaging techniques like MRI machines and in particle accelerators.

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