Biot-Savart. The equation is this:

Transcription

1 Biot-Savart When a wire carries a current, this current produces a magnetic field in the vicinity of the wire. One way of determining the strength and direction of this field is with the Law of Biot-Savart. The equation is this: dl x R db is the little contribution of magnetic field measured at point P μ 0 is the permeability of free space, x 10-7 T m A I is the current in the wire dl is the small length vector in which the current flows R is the distance between that small length of wire and where the field is measured R is the unit vector for R. It has a magnitude of one, so it doesn t affect the calculation, but does provide the second vector for the right hand rule of the cross product The above is a cross-product, so the direction of db follows from the right-hand-rule for cross products. Point the right-hand index finger up and to the right along dl. Rotate the middle finger up and to the left along R. The thumb indicates the direction of db, out of the page. Answer Webassign Question 1

2 There are only a few situations where this law can be applied without complex integrals and we ll look at two. The first is a wire bent into a half-circle of radius, R, which carries a current, I. We want to find the magnetic field strength in the center of the half-circle. If we look at a point above the circle, in the straight portion of the wire, we can see that dl points downward with the current. Also, the R will point down because this vector points from the little dl segment to where the magnetic field is being measured at the center of the half-circle. This makes dl x R = dl 1 sin0º = 0 So any small segment of wire in the straight portion above the half-circle will not contribute to the magnetic field. If you take a small segment of wire in the straight portion below the half-circle, again dl points down, but here R points upwards towards the center of the circle So dl x R = dl 1 sin180º = 0 Again, no contribution But now take a small segment in the half-circle.

3 Now dl x R = dl 1 sin90º = dl So Biot-Savart simplifies to dl And dl = μ 0 I (πr) because πr is the sum of all the little steps around a half circle 4R Answer Webassign Question 2 Answer Webassign Question 3 The direction comes from the right hand rule. If you point your index finger along with dl and then point your middle finger with R, your thumb will point out of the page with db. This will be true for all small wire segments, making the total field also out of the paper. A shortcut for this is to curl your right-hand fingers with the circulating current and your thumb will point outward with the magnetic field.

4 Now take a circular loop of radius a carrying a current I. A line is drawn from the center of that circle to the right and on that line sits a point P, a distance b from the center of the circle. When standing at P, the current is seen to circulate clockwise. What is the strength and direction of the magnetic field at P? Let s look at the small current segment at the top of the loop. By the right hand rule, point your index finger into the page with the current flow, middle finger from that segment towards P (along R ) and you will find db points down and left, perpendicular to R. For the small current segment at the bottom of the loop, point your index finger out of the page with the current flow, middle finger from that segment towards P (along R ) and you will find db points up and left, perpendicular to R. The vertical components add to zero, leaving only the horizontal components, which are dbsinθ. You can see it is the sine function if I add a few more lines to the diagram: Now if we go back to Biot-Savart, we can see that dl x R = dl 1 sin90º = dl. The angle is 90º because dl is along the z-axis and R is in the xy-plane. Also, the distance R = a 2 + b 2

5 So we have db x = μ 0 I dl a a 2 +b2 sinθ where sinθ = a 2 + b 2 Integrating is just taking the sum of all the little steps of dl, which is the circumference of the circle, 2π a. a 2 2 (a 2 +b 2 ) 3 2 Answer Webassign Question 4 Answer Webassign Question 5 Also, the direction of the field agrees with the right hand short cut of curling your fingers with the current circulation and seeing which way your thumb points. Appendix: What is the magnetic field at some distance, y, from a very long, straight, current-carrying wire? There is an easy way to find this with what is called Ampere s law. Here is the same result using Biot- Savart: dl x R We can replace dl with dx. And because the cross product uses the sin of the angle between dl and R, it will be the cos of the angle in the diagram. These substitutions produce: dx cosθ

6 Now, if you zoom-in on the little box around dl, it looks like this: So we can write dx as Rdθ and the full equation becomes: cosθ R dθ = dθ R Going back to the first diagram, R = y, so cosθ cosθ dθ y Integrating between the boundaries of - π to π (because angles are measures from the vertical) yields: 2 2 y cosθ dθ = μ 0 I y sinθ π/2 π/2 = μ 0 I 2πy