Physics 1402: Lecture 17 Today s Agenda

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Physics 1402: Lecture 17 Today s Agenda Announcements: Midterm 1 distributed today Homework 05 due Friday Magnetism Trajectory in Constant B Field Suppose charge q enters B field with velocity v as

1 Physics 1402: Lecture 17 Today s Agenda Announcements: Midterm 1 distributed today Homework 05 due Friday Magnetism Trajectory in Constant B Field Suppose charge q enters B field with velocity v as shown below. (v B) What will be the path q follows? v F v F q Force is always to velocity and B. What is path? Path will be circle. F will be the centripetal force needed to keep the charge in its circular orbit. Calculate : B 1

2 adius of Circular Orbit Lorentz force: centripetal acc: Newton's 2nd Law: v F v F q B This is an important result, with useful eperimental consequences! atio of charge to mass for an electron e - 1) Turn on electron gun 2) Turn on magnetic field B ΔV gun 3) Calculate B net week; for now consider it a measurement 4) earrange in terms of measured values, V, and B & 2

3 Lawrence's nsight " cancels " We just derived the radius of curvature of the trajectory of a charged particle in a constant magnetic field. E.O. Lawrence realized in 1929 an important feature of this equation which became the basis for his invention of the cyclotron. ewrite in terms of angular velocity ω! does indeed cancel in above eqn. So What?? The angular velocity is independent of!! Therefore the time for one revolution is independent of the particle's energy! We can write for the period, T=2π/ω or T = 2πm/qB This is the basis for building a cyclotron. The Hall Effect d l v d qe H - F c c B Force balance Hall voltage generated across the conductor B a Using the relation between drift velocity and current we can write: 3

4 The Laws of Biot-Savart & Ampere r d P dl Calculation of Electric Field Two ways to calculate the Electric Field: Coulomb's Law: "Brute force" Gauss' Law "High symmetry" What are the analogous equations for the Magnetic Field? 4

5 Calculation of Magnetic Field Two ways to calculate the Magnetic Field: Biot-Savart Law: "Brute force" Ampere's Law "High symmetry" These are the analogous equations for the Magnetic Field! Biot-Savart Law bits and pieces dl r A X db B in units of Tesla (T) µ 0 = 4π X 10-7 T m /A So, the magnetic field circulates around the wire 5

6 Magnetic Field of Straight Wire Calculate field at point P using Biot-Savart Law: Which way is B? r d P ewrite in terms of,:, Magnetic Field of Straight Wire r d P 6

7 Lecture 17, ACT 1 have two wires, labeled 1 and 2, carrying equal current, into the page. We know that wire 1 produces a magnetic field, and that wire 2 has moving charges. What is the force on wire 2 from wire 1? Wire 1 (a) Force to the right (b) Force to the left (c) Force = 0 X F Wire 2 X B 7

8 Force between two conductors Force on wire 2 due to B at wire 1: Force on wire 2 due to B at wire 1: Total force between wires 1 and 2: Direction: attractive for 1, 2 same direction repulsive for 1, 2 opposite direction Circular Loop Circular loop of radius carries current i. Calculate B along the ais of the loop: Magnitude of db from element dl: z r r db db z What is the direction of the field? Symmetry B in z-direction. 8

9 Circular Loop z r r db db z Note the form the field takes for z>>: Epressed in terms of the magnetic moment: note the typical dipole field behavior! Circular Loop B z 1 z z 9

10 Lecture 17, ACT 2 Equal currents flow in identical circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction. What is the magnetic field B z (A) at point A, the midpoint between the two loops? (a) B z (A) < 0 (b) B z (A) = 0 (c) B z (A) > 0 Lecture 17, ACT 3 Equal currents flow in identical circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction. What is the magnetic field B z (B) at point B, just to the right of the right loop? (a) B z (B) < 0 (b) B z (B) = 0 (c) B z (B) > 0 10

Magnetic Field of Straight Wire Calculate field at distance from wire using Ampere's Law: Choose loop to be circle of radius centered on the wire in a plane to wire. Why?

11 Magnetic Field of Straight Wire Calculate field at distance from wire using Ampere's Law: Choose loop to be circle of radius centered on the wire in a plane to wire. Why?» Magnitude of B is constant (fct of only)» Direction of B is parallel to the path. Evaluate line integral in Ampere s Law: Current enclosed by path = Apply Ampere s Law: dl Ampere's Law simplifies the calculation thanks to symmetry of the current! ( aial/cylindrical ) 11

12 B Field inside a Long Wire? What is the B field at a distance, with <a (a: radius of wire)? adius a Choose loop to be circle of radius, whose edges are inside the wire. Why?» Left Hand Side is same as before. Current enclosed by path = J Area of Loop Apply Ampere s Law: 12

nside the wire: (r < a) eview: B Field of a Long Wire a Outside

infinite straight wire in the +z direction as shown.

13 nside the wire: (r < a) eview: B Field of a Long Wire a Outside the wire: (r>a) B r Lecture 17, ACT 4 A current flows in an infinite straight wire in the +z direction as shown. A 2A concentric infinite cylinder of radius carries current in the -z direction. What is the magnetic field B (a) at point a, just outside the cylinder as shown? (a) B (a) < 0 (b) B (a) = 0 (c) B (a) > 0 13

2B Lecture 17, ACT 4 A current flows in an infinite straight wire in the +z direction as shown. A concentric infinite cylinder of radius carries current in the -z direction.

14 2B Lecture 17, ACT 4 A current flows in an infinite straight wire in the +z direction as shown. A concentric infinite cylinder of radius carries current in the -z direction. What is the magnetic field B (b) at point b, just inside the cylinder as shown? (a) B (b) < 0 (b) B (b) = 0 (c) B (b) > 0 B Field of a Solenoid A constant magnetic field can (in principle) be produced by an sheet of current. n practice, however, a constant magnetic field is often produced by a solenoid. A solenoid is defined by a current flowing through a wire which is wrapped n turns per unit length on a cylinder of radius a and length L. L a f a << L, the B field is to first order contained within the solenoid, in the aial direction, and of constant magnitude. n this limit, we can calculate the field using Ampere's Law. 14

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B Field of a Solenoid To calculate the B field of the solenoid using Ampere's Law, we need to justify the claim that the B field is 0 outside the solenoid.

16 B Field of a Solenoid To calculate the B field of the solenoid using Ampere's Law, we need to justify the claim that the B field is 0 outside the solenoid. To do this, view the solenoid from the side as 2 current sheets. The fields are in the same direction in the region between the sheets (inside the solenoid) and cancel outside the sheets (outside the solenoid). Draw square path of side w: (n: number of turns per unit length) 16

17 Toroid Toroid defined by N total turns with current i. B=0 outside toroid! (Consider integrating B on circle outside toroid) To find B inside, consider circle of radius r, centered at the center of the toroid. r B Apply Ampere s Law: Magnetic Flu Define the flu of the magnetic field through a surface (closed or open) from: ds B B Gauss s Law in Magnetism 17

Magnetism in Matter When a substance is placed in an eternal magnetic field B o, the total magnetic field B is a combination of B o and field due to magnetic moments (Magnetization; M): B = B o + µ o

18 Magnetism in Matter When a substance is placed in an eternal magnetic field B o, the total magnetic field B is a combination of B o and field due to magnetic moments (Magnetization; M): B = B o + µ o M = µ o (H +M) = µ o (H + χ H) = µ o (1+χ) H» where H is magnetic field strength χ is magnetic susceptibility Alternatively, total magnetic field B can be epressed as: B = µ m H» where µ m is magnetic permeability» µ m = µ o (1 + χ ) All the matter can be classified in terms of their response to applied magnetic field: Paramagnets µ m > µ o Diamagnets µ m < µ o Ferromagnets µ m >>> µ o 18

Physics 1402: Lecture 18 Today s Agenda

Physics 1402: Lecture 18 Today s Agenda Announcements: Midterm 1 distributed available Homework 05 due Friday Magnetism Calculation of Magnetic Field Two ways to calculate the Magnetic Field: iot-savart