The wire semicircles shown in Fig. P 28.62 have radii a and b. Calculate the net magnetic field

step 1 Okay, we're looking at question 28 .66. We have two wire semicircles that are connected here to

The wire semicircles shown in Fig. P 28.62 have radii a and...

Key Concepts

Right-Hand Rule

The right-hand rule is a practical tool used to determine the direction of the magnetic field generated by a current carrying conductor. By pointing the thumb in the direction of the conventional current and curling the fingers, the direction in which the magnetic field circulates is revealed. This rule ensures that the vector nature of magnetic fields is properly accounted for in calculations and that the orientation of the net field is correctly determined.

Superposition Principle

The superposition principle states that when multiple magnetic fields are present, the net magnetic field is the vector sum of the individual fields produced by each current segment. This concept is crucial in problems where the magnetic field is produced by separate portions of a current-carrying conductor, allowing one to calculate the overall effect by summing the contributions from each segment independently.

Magnetic Field due to a Circular Arc

This concept concerns the derivation of the magnetic field at the center of a circular arc, which is a segment of a full circle. For a full circular loop, the magnetic field is determined by a well-known formula; for arcs such as semicircles, the field is obtained by taking an appropriate fraction of the complete loop’s field. Understanding this relationship helps in simplifying and computing the resultant field for non-complete loops.

Biot–Savart Law

The Biot–Savart Law is the fundamental principle used to calculate the magnetic field generated by a current element. It relates the current, the displacement vector from the current element to the point of interest, and the current's geometry. This law is integral in analyzing magnetic fields produced by various current configurations, including wires shaped into arcs or loops.

Transcript

00:01 Okay, we're looking at question 28 .66.

00:04 We have two wire semicircles that are connected here to make a current loop.

00:12 And we're trying to find the net magnetic field that the current and the wires produce at this point.

00:18 So this point is a distance a from this inner semicircle and distance b from the outer semicircle.

00:25 Now the first thing to notice, well, let's take a look at the b .s.

00:29 Of r law, which is what we're going to need to use to calculate this.

00:32 And the essential component is this dl cross r.

00:36 So dl is going to be along the length of this wire.

00:40 And then r will be the vector pointing from the point to the point on the wire.

00:45 And if we take a look at this, notice that these little horizontal components are not going to contribute anything to the magnetic field here.

00:54 Because the cross product, between the between dl, which is along the plus or minus x direction, is parallel to r, which is also along the plus or minus x direction.

01:06 So these two components, we can just totally ignore because they're not going to contribute anything to the field here.

01:14 So let's start by just going ahead and calculating the magnetic field from this first part.

01:26 So let's just basically copy this over here.

01:32 We're gonna get, let's say that we're defining current to be positive in this direction to the left along here.

01:41 So we'll get positive i and then dl cross r is really just going to be dl times r because if we notice since this is a semicircle, it's not drawn very well, but since it's a semicircle, it's going to be perpendicular to the tangent, which is dl, is going to be perpendicular to r, which is going to be the vector pointing to the center of the circle at every point along this.

02:07 So dl and r are going to be perpendicular for all points here.

02:12 And that means that the cross product is just going to be equal to the regular product of their magnitudes.

02:17 So we're going to have dl times r here, and then a factor of r cubed on the bottom.

02:22 So right away, we can just cancel out one factor of r, leaving us with r squared.

02:28 And notice that r here, since we're looking at this inner one, is just going to be equal to a.

02:35 And now these are all things that we either, we don't necessarily know the values, but we're asked, we're told what i is, and we're told we have a.

02:43 So now we just need this dl.

02:45 So i'm going to pull this factor of dl out because what we're going to need to do is integrate over this expression.

02:55 So i'm going to just hold off on that for a minute because now what we want to do is add to the, we want to add the component of the magnetic field from this second outer part.

03:14 So the outer part is just going to be the exact same thing except what we have.

03:21 Instead, instead of a here, this is just going to be b.

03:26 And we know that, again, regardless of which direction we define the current in, we'll decide the direction of the magnetic field in a minute.

03:34 We know that it's going to give an opposite contribution as the other one.

03:38 So i'm just going to say that this i is going to be negative.

03:41 So let's call this minus, because we know these aren't going to just add total.

03:46 We know that one is going to be bigger than the other.

03:48 And again, we'll figure that out in just a minute.

03:51 But what we need to do at this point is integrate this to get b...

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