(II) A segment of wire of length d carries a current I as shown in Fig. 49 . (a) Show that for p

(II) A segment of wire of length d carries a current I as...

Key Concepts

Biot-Savart Law

The Biot?Savart law is a fundamental principle used to calculate the magnetic field produced by a moving charge or a current element. It relates the magnetic field at a point in space to the magnitude, direction, and distance of the current elements, requiring an integration over the current distribution to account for the contributions from each segment. This law is central to understanding how fields are generated by current-carrying wires and is the starting point for analyzing the magnetic field in many geometries.

Symmetry in Magnetic Field Problems

Symmetry is a powerful tool in simplifying electromagnetic problems, especially when assessing the contributions to the magnetic field from a current-carrying wire. In the context of the problem, symmetry arguments help show that certain components of the magnetic field cancel each other. When computing the net field along the axis of the wire, contributions from current elements on opposite sides of a point may cancel due to their equal magnitude and opposite direction, resulting in a zero net field.

Integration Over Finite Current Elements

When dealing with finite segments of current-carrying wires, the magnetic field is determined by integrating the contributions from each infinitesimal current element along the wire. This integration, guided by the Biot-Savart law, involves setting up appropriate coordinate systems and geometric relations to account for the distance and orientation of each segment relative to the observation point. The resulting expressions capture the cumulative effect of the current segment on the magnetic field at various points in space.

Geometric Considerations in Electromagnetism

The geometry of the system—including the orientation of the wire and the position of the point where the field is being calculated—plays a crucial role in electromagnetic analysis. For instance, calculating the magnetic field along different axes of a current-carrying wire involves considering distances, angles, and symmetry properties. Correctly setting up the coordinate system and understanding these geometric relationships are key to applying the Biot-Savart law and successfully evaluating the magnetic field.

Transcript

00:01 Our question says to use figure 49 as reference, and it wants us to show that for points along the positive x -axis, the axis of the wire, the magnetic field b is zero.

00:11 Okay, so we can go ahead and do that, and we'll indicate this as part a.

00:19 So consider a small element on the rod.

00:24 Small element of change, we'll call this element dl, at a distance r from the point q, as shown in the figure.

00:31 So from the biaz -s -sav -art law, the magnetic field at point q is going to be equal to, well, of course it's b, the magnetic field, is going to be equal to, mu -not over 4 pi, times the integral of the current i times dl, l is that infinitesimal length change that we were talking about, cross vector r divided by r cubed.

01:07 Cubed, okay? and again, r is the distance that we're considering from the point q on the figure.

01:21 And so since the point q is along the length of the wire, then dl and r are parallel.

01:27 So we can tell that dl is, we can say parallel.

01:32 This is the symbol for parallel with r.

01:35 Since that's true, dl cross r is zero, because the cross product of two things that are parallel, is zero.

01:49 Well, you plug zero into that integral, and anything times zero is zero, so therefore the magnetic field b is zero.

01:57 And that is our proof.

01:58 We can box that in as our solution for a.

02:02 Part b says to determine a formula for the field at points along the y -axis, such as point p.

02:11 Okay, so consider the small element d, dx on the rod.

02:18 So we're going to let the distance between the element dl and the point p, b little r.

02:24 So from the right -hand rule, the magnetic field at point p due to the current in the wire is out of the page as shown.

02:29 So we're going to call out of the page the positive k -hat direction.

02:33 So what do we have? we have b.

02:37 We'll indicate this as part b.

02:39 We have the magnetic field b.

02:44 Sorry, the cursor got stuck there.

02:45 Let's rewrite that.

02:47 There we go.

02:49 Is equal to, we'll pull everything out of the integral that's a constant here.

02:52 So we have mu not times i divided by 4 pi.

03:03 And then the integral here is dx.

03:07 That's what we're integrating with respect to times sine theta over little r squared.

03:19 And then this is all going in the out of the page direction or the k had direction.

03:24 So we're going to use some trigonometric identities to simplify this a little bit.

03:27 So from looking at the graph, we have r is equal to y over sine theta.

03:32 Because sine theta is equal to y over r or opposite over hypotenuse...

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