Magnetic Field due to long wire?

In summary: If s = 0, then the two elements are at the same distance from P. If s differs by 10.5% from 0, then the dB produced by the two elements at s = 0 will be 10.5% of the dB produced by the two elements at s = 10.5%.In summary, the distance to the element making the greatest contribution to the magnetic field at point P is 0.
  • #1
Destroxia
204
7

Homework Statement



In the figure, point P is at perpendicular distance R = 2.32 cm from a very long straight wire carrying a current. The magnetic field B set up at point P is due to contributions from all the identical current-length elements ids along the wire. What is the distance s to the element making (a) the greatest contribution to field B and (b)10.5% of the greatest contribution?

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Homework Equations



Biot- Savart Law = u0/4pi * (ids*sin(theta))/(r^2)

The Attempt at a Solution



I tried to set up an integration using the biot savart law and came up with something along the lines of,

u0*i*R/(4pi) = ∫(-inf/inf) ds/(R^2+ s^2)^(3/2)

I solved (a) simply because I figured the strongest point of the magnetic field must be at the point closes to point P, the answer was 0, therefore.

I don't understand (b), if 0 is the greatest contribution, 10.5% of the greatest contribution would also be 0... Yet it is not.
 
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  • #2
OK for (a).

Let dBmax be the contribution from the current element that contributes the most. Part (b) is asking for the distance s for an element that contributes 10.5% of dBmax.
 
  • #3
TSny said:
OK for (a).

Let dBmax be the contribution from the current element that contributes the most. Part (b) is asking for the distance s for an element that contributes 10.5% of dBmax.

So, essentially, I'm just finding the total B, taking 10.5% of that, and solving for the distance in the biot-savart law?
 
  • #4
No, you do not need to find the total B. You are just comparing the contributions dB from individual segments (elements). In part (a) you considered the segment that produces the maximum dB. As you noted, this is the segment at s = 0. In (b) you want to find a segment that contributes a dB which is 10.5% of the amount contributed by the segment at s = 0.
 
  • #5
TSny said:
No, you do not need to find the total B. You are just comparing the contributions dB from individual segments (elements). In part (a) you considered the segment that produces the maximum dB. As you noted, this is the segment at s = 0. In (b) you want to find a segment that contributes a dB which is 10.5% of the amount contributed by the segment at s = 0.

I guess I just meant, do I need to find the B at s = 0 then? If I don't, how would I find 10.5% of s = 0, If I don't know the value for that point?
 
  • #6
You do not need to find a numerical value for the infinitesimal magnetic field, dB, produced at P by the element Ids located at s = 0. That would require knowing I and ds.

Think about the ratio of dB produced by two different current elements located at different values of s.
 

Related to Magnetic Field due to long wire?

1. What is the formula for calculating the magnetic field due to a long wire?

The formula for calculating the magnetic field due to a long wire is given by B = (μ0 * I) / (2 * π * r), where B is the magnetic field strength, μ0 is the permeability of free space, I is the current in the wire, and r is the distance from the wire.

2. How does the strength of the magnetic field vary with distance from the wire?

The strength of the magnetic field decreases as the distance from the wire increases. This is because the field lines spread out as they move away from the wire, resulting in a lower field strength at a greater distance.

3. What factors affect the strength of the magnetic field due to a long wire?

The strength of the magnetic field is affected by the current in the wire, the distance from the wire, and the permeability of the material surrounding the wire. Additionally, the angle between the wire and the observer's point also affects the field strength.

4. Can the direction of the magnetic field be changed?

Yes, the direction of the magnetic field can be changed by changing the direction of the current in the wire. The field lines always form concentric circles around the wire, with the direction of the field determined by the direction of the current.

5. How is the magnetic field due to a long wire related to the right-hand rule?

The right-hand rule can be used to determine the direction of the magnetic field around a long wire. If the thumb of the right hand points in the direction of the current, the fingers will curl in the direction of the field lines. This rule also applies to other situations involving magnetic fields, such as those created by magnets or current-carrying loops.

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