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kostoglotov

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This is a problem from a section on Line Integrals in my Calculus Textbook, I haven't studied any physics relating to E&M yet, and the solutions manual only gives solutions for odd numbered problems. Sorry, if I'm posting in the wrong forum, I hope I'm not.

A steady current [itex]I[/itex] in a long wire produces a magnetic field [itex]\vec{B}[/itex] that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the wire.

Ampere's Law relates the electric current to its magnetic effects and states that:

[tex]\int_C \vec{B} \cdot d\vec{r} = \mu_0I[/tex]

where [itex]I[/itex] is the net current that passes through any surface bounded by a closed curve C, and [itex]\mu_0[/itex] is a constant called the permeability of free space. By taking C to be a circle with radius r, show that the magnitude [itex]B = |\vec{B}|[/itex] of the magnetic field at a distance [itex]r[/itex] from the center of the wire is

[tex]B = \frac{\mu_0I}{2\pi r}[/tex]

[tex]\vec{A}\cdot \vec{B} = |\vec{A}||\vec{B}|\cos{\theta}[/tex]

I also found this website that gives a different mathematical form for Ampere's Law

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq2.html#c4

[tex]\oint B ds = \mu_0I + \frac{1}{c^2}\int E\cdot dA[/tex]

But I don't know what that second term is, so I am assuming that the form of Ampere's Law given me by the textbook is specific to closed

So [itex]\int_C \vec{B}\cdot d\vec{r}=\int_C \vec{B} \cdot \vec{T} ds[/itex] where [itex]\vec{T}[/itex] is a unit vector and so [itex]|\vec{T}| = 1[/itex]

Also, [itex]ds = r dt[/itex] since the [itex]\vec{r} = \langle r\cos(t),r\sin(t)\rangle[/itex]

Then, [itex]\vec{B} \cdot \vec{T} = |\vec{B}||\vec{T}|\cos{\theta}[/itex]. Obviously our unit tangent vector magnitude just becomes one, but here I also need to argue that [itex]\theta=0[/itex].

Since it states that the magnetic field will be tangent to a circle at radius r from the wire, I feel comfortable stating that theta will be zero on the circular path.

Here, one continues [itex]\int_C \vec{B} \cdot \vec{T} ds = \int_C |\vec{B}| ds = \mu_0I[/itex], provided C is a circle of radius r in the plane perpendicular to the wire/source and with the wire/source at the center of the circle.

So [itex]\int_C |\vec{B}| ds = \int_0^{2\pi} |\vec{B}| r dt = |\vec{B}|r[t]_0^{2\pi}[/itex], but now I need to argue that [itex]|\vec{B}|[/itex] and [itex]r[/itex] are constants. This is no problem for [itex]r[/itex] as it is assumed to be a constant from the beginning of the exercise, but I am not 100% comfortable arguing that [itex]|\vec{B}|[/itex] is a constant in this case.

I could argue that [itex]|\vec{B}|[/itex] is a constant because it is a function of [itex]r[/itex], and it states that in the setup to the problem itself, so I have no big issue with this intuitively. If we assume something to be so, then so be it...but...

Is there anything in this mathematical statement of Ampere's Law

[tex]\int_C \vec{B} \cdot d\vec{r} = \mu_0I[/tex]

that tells us for certain that [itex]|\vec{B}|[/itex] is a function of [itex]r[/itex] and that if [itex]r[/itex] is constant then so is [itex]|\vec{B}|[/itex]? After all, it is a line integral that equals a constant value.

If a line integral is a constant value, is it generally then the case that the magnitude of the vector field is a function only of the path through that field? Or only for closed paths? Is this what they mean by a conservative field? Or am I just veering completely into nonsense?

1. Homework Statement1. Homework Statement

A steady current [itex]I[/itex] in a long wire produces a magnetic field [itex]\vec{B}[/itex] that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the wire.

Ampere's Law relates the electric current to its magnetic effects and states that:

[tex]\int_C \vec{B} \cdot d\vec{r} = \mu_0I[/tex]

where [itex]I[/itex] is the net current that passes through any surface bounded by a closed curve C, and [itex]\mu_0[/itex] is a constant called the permeability of free space. By taking C to be a circle with radius r, show that the magnitude [itex]B = |\vec{B}|[/itex] of the magnetic field at a distance [itex]r[/itex] from the center of the wire is

[tex]B = \frac{\mu_0I}{2\pi r}[/tex]

## Homework Equations

[tex]\vec{A}\cdot \vec{B} = |\vec{A}||\vec{B}|\cos{\theta}[/tex]

I also found this website that gives a different mathematical form for Ampere's Law

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq2.html#c4

[tex]\oint B ds = \mu_0I + \frac{1}{c^2}\int E\cdot dA[/tex]

But I don't know what that second term is, so I am assuming that the form of Ampere's Law given me by the textbook is specific to closed

**circular**paths around the wire. Is this correct? What is this second term all about?## The Attempt at a Solution

So [itex]\int_C \vec{B}\cdot d\vec{r}=\int_C \vec{B} \cdot \vec{T} ds[/itex] where [itex]\vec{T}[/itex] is a unit vector and so [itex]|\vec{T}| = 1[/itex]

Also, [itex]ds = r dt[/itex] since the [itex]\vec{r} = \langle r\cos(t),r\sin(t)\rangle[/itex]

Then, [itex]\vec{B} \cdot \vec{T} = |\vec{B}||\vec{T}|\cos{\theta}[/itex]. Obviously our unit tangent vector magnitude just becomes one, but here I also need to argue that [itex]\theta=0[/itex].

Since it states that the magnetic field will be tangent to a circle at radius r from the wire, I feel comfortable stating that theta will be zero on the circular path.

Here, one continues [itex]\int_C \vec{B} \cdot \vec{T} ds = \int_C |\vec{B}| ds = \mu_0I[/itex], provided C is a circle of radius r in the plane perpendicular to the wire/source and with the wire/source at the center of the circle.

So [itex]\int_C |\vec{B}| ds = \int_0^{2\pi} |\vec{B}| r dt = |\vec{B}|r[t]_0^{2\pi}[/itex], but now I need to argue that [itex]|\vec{B}|[/itex] and [itex]r[/itex] are constants. This is no problem for [itex]r[/itex] as it is assumed to be a constant from the beginning of the exercise, but I am not 100% comfortable arguing that [itex]|\vec{B}|[/itex] is a constant in this case.

I could argue that [itex]|\vec{B}|[/itex] is a constant because it is a function of [itex]r[/itex], and it states that in the setup to the problem itself, so I have no big issue with this intuitively. If we assume something to be so, then so be it...but...

Is there anything in this mathematical statement of Ampere's Law

[tex]\int_C \vec{B} \cdot d\vec{r} = \mu_0I[/tex]

that tells us for certain that [itex]|\vec{B}|[/itex] is a function of [itex]r[/itex] and that if [itex]r[/itex] is constant then so is [itex]|\vec{B}|[/itex]? After all, it is a line integral that equals a constant value.

If a line integral is a constant value, is it generally then the case that the magnitude of the vector field is a function only of the path through that field? Or only for closed paths? Is this what they mean by a conservative field? Or am I just veering completely into nonsense?

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