CALCULATING MAGNETIC FIELDS & THE BIOT-SAVART LAW. Purdue University Physics 241 Lecture 15 Brendan Sullivan

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1 CALCULATING MAGNETIC FIELDS & THE BIOT-SAVAT LAW Purdue University Physics 41 Lecture 15 Brendan Sullivan

2 Introduction Brendan Sullivan, PHYS89, Office Hours: By Appointment Just stop by my office any time you have a question ight after class or recitation is a pretty great time for this I ll run lecture nearly the same the first half Quizzes scored the same, etc. Will occasionally start class with a reading quiz

3 Overview Magnetic field for moving point charges Magnetic field for a current (collection of moving point charges) Current Loops Solenoids Straight Wires Combinations of Straight Wires (Superposition)

4 Magnetic Fields are Produced by Moving Charges We know magnetic fields interact with moving charges (and currents) E.g. Cyclotron motion Now we want to be able to explain where magnetic fields come from Short answer: A moving charge produces a magnetic field F qv B x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x + x x x x x x x x x x x x x x x x x x x x x x x x x x

5 Magnetic Fields Are Generated By Moving Charges Magnetic field for a point charge: B qv rˆ 4 r

6 Magnetic Fields Are Generated By Moving Charges Magnetic field for a point charge: The magnetic field (B) is given by Divided by the distance squared Some constant B The particle s charge and velocity qv rˆ 4 r Crossed into r-hat, a unit vector that points from the charge to where we re trying to find the field at µ is known as the permeability of free space µ = 4πx1-7 T m/a = 4πx1-7 N/A

7 Consequences of a Moving Charge B qv rˆ 4 r The cross product picks out the perpendicular part B = along the direction of travel (red line) B is maximal perpendicular to direction of travel (blue line) B dies off as 1/r B = at r = For a worked example of how to calculate B, see example 7-1 in text (pg. 918)

8 Quiz Question 1 An electron is traveling in the +x direction. As is passes through the origin, which direction if the field at r = <5, 5, >? a) +z b) z c) +y d) y e) Trick question! B =! (don t pick this one)

9 Currents are a collection of moving charges, which create a B field arely do we see single charges in the real world; we see currents qv is like a current If we consider a tiny bit of wire (dl) we can get the infinitesimal field db If we know db, we can use superposition to figure out the total B field at some point Integrate, just like finding E for charge distributions B db qv rˆ 4 r Differential Id rˆ 4 r We call this the Biot-Savart Law

10 The Biot-Savart Law db Id rˆ 4 r Or (easier to think about, Id rˆ db but not technically correct) 4 r dl points in the direction of the current Helpful to think of I as the vector Everything else works the same as for a point charge I, dl rˆ x

11 Biot-Savart Law for a loop of charge Let s find the B field at some point x on the symmetry axis for the ring shown below. (Done on DocCam take notes)

12 Biot-Savart Law for a loop of charge Let s find the B field at some point x on the symmetry axis for the ring shown below. (Done on DocCam take notes) We found the field pointed only in the symmetry (x) direction: B 3 ( x I ),, Very far away (dipole limit, x>>) we recover that B x I 3 x

13 Left: Electric field for electric dipole; Middle: B for bar magnet; ight: B for current loop (dipole) Current loops are magnetic dipoles This field is the analog to an electric dipole E x 1 4 p 3 x B x I I x 4 x 4 x

14 B at the center of a current loop For a full loop, we merely set x = I B x =,, 3 ( x ) B x I In general, we can just take a fraction of that if we have only part of a current loop Qualitatively: Each section of the loop contributes equally We can also apply superposition to loops

15 Quiz Question BAD QUESTION Two loops are centered on the x-axis as shown. Each has a clockwise current I as viewed from the +x axis. Determine B at the origin. x y d d B x I B ˆ x I B ˆ x I B ˆ x I B ˆ a) b) c) d) e)

16 Supplement: Quiz Question x o x o x d d Side view: o s are current out of the page, x s are current into the page I should ve mentioned the rings had radius Both rings have fields going to the right (this applies to both sides of the ring; you can prove this if you get bored the field lines offer a clear view of this) Superposition Add the two fields

17 Supplement: Quiz Question In the figure, the blue arrow is the left loop s field and the orange is the right (at the origin) Add the two fields: The answer should be: d B x d ( x ( x I I ) 3/ ) 3/ ( x I ) 3/

18 Solenoids are just a lot of current loops We can get the magnetic field on the axis of a solenoid by adding up all the rings contributions! Done on DocCam

19 The magnetic field is essentially uniform in a very long solenoid z 1 z z dz B z z B z 1 ni( ( z z z 1 ) z 1 ( z z z z ) ) Important: n here stands for turn density (turns / m), not the total number of turns! Very Long, for far from ends B z ni

20 B due to a straight segment of wire Start from Biot-Savart Law Express everything in terms of θ, noting that x has a dependence on θ Then integrate along θ Done on DocCam

21 B due to a straight segment of wire The field wraps in circles around the wire, we can use the right hand rule to determine CW or CCW I B B (sin( ) sin( 1)) 4 Infinitely Long Wire I B B Note: we can use superposition with wires as well

22 Quiz Question 3 Two infinite wires are carrying current I as shown below. What is the magnetic field along the midline (dashed line?) I a) b) c) d) e) B B I d B I d B I d B I d Into the page Out of the page Into the page Out of the page d d I

23 Two currents near each other have a pinching effect I 1 I F L I1I Consider two nearby currents, I 1 and I I 1 creates a magnetic field where I is located Into the page, as pictured Now, I is a current in a magnetic field, which creates a force on the wire To the left, as pictured Vice-versa for I 1, equal and opposite Since infinite wires, we find force per unit length Derived on DocCam it s just the product of the current and magnetic field!

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