Experiments show that a steady current I in a long wire...

Experiments show that a steady current $I$ in a long wire produces a magnetic field $\mathbf{B}$ that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the wire (as in the figure at the right). Ampere's Law relates the electric current to its magnetic effects and states that $$ \int_{C} \mathbf{B} \cdot d \mathbf{r}=\mu_{0} I$$ where $I$ is the net current that passes through any surface bounded by a closed curve $C,$ and $\mu_{0}$ is a constant called the permeability of free space. By taking $C$ to be a circle with radius $r,$ show that the magnitude $B=|\mathbf{B}|$ of the magnetic field at a distance $r$ from the center of the wire is $$B=\frac{\mu_{0} I}{2 \pi r}$$

Experiments show that a steady current $I$ in a long wire produces a magnetic field $\mathbf{B}$ that is tangent to any circle that lies in the plane perpendicular to the wire and whose
center is the axis of the wire (as in the figure at the right). Ampere's Law relates the electric current to its magnetic effects and states that
$$ \int_{C} \mathbf{B} \cdot d \mathbf{r}=\mu_{0} I$$
where $I$ is the net current that passes through any surface bounded by a closed curve $C,$ and $\mu_{0}$ is a constant called the permeability of free space. By taking $C$ to be a circle
with radius $r,$ show that the magnitude $B=|\mathbf{B}|$ of the magnetic field at a distance $r$ from the center of the wire is $$B=\frac{\mu_{0} I}{2 \pi r}$$

Key Concepts

Ampere's Law

Ampere's Law relates the line integral of the magnetic field around a closed loop to the net electric current passing through the surface bounded by that loop. This fundamental law in magnetism is particularly useful in situations with high symmetry, as it simplifies the calculation of the magnetic field by reducing the integral to a product of the field magnitude and the length of the chosen path.

Circular Symmetry

Circular symmetry arises when the geometry of a current distribution, such as a long straight wire, ensures that the magnetic field at any point equidistant from the wire has the same magnitude and a direction tangent to a circle centered on the wire's axis. This symmetry allows the assumption of a constant magnetic field magnitude along an appropriately chosen circular Amperian loop.

Amperian Loop

An Amperian loop is an imaginary closed path used in applying Ampere's Law. Selecting a loop that aligns with the symmetry of the current configuration simplifies the evaluation of the line integral, since the magnetic field is constant in magnitude and direction (relative to the loop) along the path, thereby facilitating the calculation of the magnetic field magnitude.

Magnetic Permeability

The magnetic permeability, denoted by ?? in free space, is a constant that quantifies the ability of the medium to support the formation of a magnetic field. In Ampere's Law, ?? serves as the proportionality factor that links the net current passing through the surface to the magnetic field produced around the current-carrying wire.

Transcript

00:01 B .dr is equal to mu .0 .i.

00:04 And we're asked to show, so wts means well, to show that b is equal to u .n .i divided by 2 .ir.

00:15 All right, and we're also given that c is x squared plus y squared is equal to r square.

00:22 So we have a circle.

00:24 And then what we do parameterize a circle, we just set x equal to r cosine, and y is equal to r sine t.

00:37 And then, so we want to change everything, so the integral along c of b .d .r, we want to write everything in terms of okay, so we take the derivative of r, which is r prime, and then we get the derivative of r cosine t is negative r s -t and the derivative of r -sign t is r -cosite.

01:03 All right, so now what we have to determine, is big b or the magnetic field? well, we have to think about this for a little bit.

01:11 So what we notice is that the magnetic field is going to be tangent to the curve c.

01:19 So basically what we have is a wire, then we have a plane, then we have circles around this plane.

01:27 And we know that the magnetic field b, just like r prime is going to be tangent to c.

01:34 So that means b is going to be a little.

01:35 The same form as r prime so our so our r prime was negative r sine t and our comma r cosy so since both r prime and b are going to be tangent to the curve c or the circle it's going to be of the same form so b is negative sign t and a comma b cosine right so now we plug everything in so we're going to plug in so now we're going to plug into integral over c of b .r sorry yeah see yeah b .dr and then we're going to rewrite it now in terms of t so it's going to be the integral from 0 to 2 pi of b .r prime all right so b is just negative b sine t comma b cosine and our r prime is going to be r -sign t, sorry, negative r -sign t, comma r -cosy.

02:40 And we take the dot product of that.

02:44 So what we get is br -sign -square -t plus br -cosin -square -t v -t...

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