A very long, cylindrical wire of radius R carries a current...

A very long, cylindrical wire of radius $R$ carries a current $I_{0}$ uniformly distributed across the cross section of the wire. Calculate the magnetic flux through a rectangle that has one side of length $W$ running down the center of the wire and another side of length $R$, as shown in Fig. P29.53 (see Exercise 29.7)

A very long, cylindrical wire of radius $R$ carries a current $I_{0}$ uniformly distributed across the cross section of the
wire. Calculate the magnetic flux through a rectangle that has one side of length $W$ running down the center of the wire and another side of length $R$, as shown in Fig. P29.53 (see Exercise 29.7)

Key Concepts

Ampere's Law

Ampere’s Law relates the circulation of a magnetic field around a closed loop to the net current passing through the loop. It is a fundamental tool in magnetostatics, particularly useful for calculating magnetic fields in situations with high symmetry such as long cylindrical wires.

Uniform Current Distribution

Uniform current distribution means that the current is spread evenly across the entire cross-sectional area of the conductor. This property simplifies calculations, as it allows the current density to be treated as a constant value, facilitating the use of Ampere's Law to derive expressions for the magnetic field.

Magnetic Field Inside a Conductor

Within a conductor carrying a uniformly distributed current, the magnetic field increases linearly with distance from the center. This behavior, derived using Ampere’s Law, stems from the fact that the enclosed current is proportional to the area within the loop being considered.

Magnetic Flux

Magnetic flux is defined as the integral of the magnetic field over a given surface area. It quantifies the amount of magnetic field passing through the area and is a key concept in electromagnetism, especially in the analysis of inductive phenomena and magnetic circuits.

Surface Integration

Surface integration involves summing contributions of a field over a continuous area to determine total effects, such as magnetic flux. This method is essential when dealing with spatially varying fields, as it allows for the accurate calculation of cumulative quantities over areas with non-uniform field strengths.

Transcript

00:01 Okay, so we're doing chapter 29, problem 55.

00:05 So in this problem, we have a big wire here, and it's carrying a current i -knot uniformly across the whole wire, and it has a radius big r.

00:19 So what we want to figure out here is we want to calculate the magnetic flux through a particular rectangle running through the center of the wire.

00:29 So to draw that, let's look at a top -down view of the wire, assume this is like infinitely long.

00:37 Here is the center axis of the wire.

00:41 So this being the radius, big r.

00:44 So what we're going to do is we have a rectangle of width big r and the length big w.

00:53 This is along there.

00:56 And we want to figure out what is flux through that just that rectangle.

01:05 Okay.

01:06 So to first do this, we know that the magnetic field depends on a while we're inside the wire and we can look back to the last chapter actually at example 288 and we'll see that the magnetic field inside of a wire dependent on r such as u not i over 2 pi big r squared linearly in r so now if we know that the magnetic flux is changing as we go we can write the differential out as b of r times the area is also changing as we go through this and this is w times the r to illustrate what this is actually meaning let's uh draw this wire again so here's our square here it's big r so what we have is imagine we're just cutting out this tiny little rectangle that is r away and of width dr and this is what we're left with.

02:20 So it's width d r times the length of w to get that area times the magnetic field b of r...

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