Chapter 33. The Magnetic Field - Welcome | Department of ... · Magnetostatics ∫B•ds=µ o I...

Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Chapter 33. The Magnetic Field Digital information is stored on a hard disk as microscopic patches of magnetism. Just what is magnetism? How are magnetic fields created? What are their properties? These are the questions we will address. Chapter Goal: To learn how to calculate and use the magnetic field.


What is the shape of the trajectory that a charged particle follows in a uniform magnetic field?

A. Helix B.  Parabola C.  Circle D. Ellipse E.  Hyperbola


What is the shape of the trajectory that a charged particle follows in a uniform magnetic field?

A.  Helix B.  Parabola C.  Circle D. Ellipse E.  Hyperbola


The magnetic field of a straight, current-carrying wire is

A.  parallel to the wire. B.  inside the wire. C.  perpendicular to the wire. D.  around the wire. E.  zero.


The magnetic field of a straight, current-carrying wire is

A.  parallel to the wire. B.  inside the wire. C.  perpendicular to the wire. D.   around the wire. E.  zero.


http://solar.gmu.edu/teaching/2008_CSI769/solar_magnetic_field.jpg

The Earth and Sun are magnetic


Connections to current A magnetic field can be sensed with a magnetic material (compass) and is associated with a current. The earth’s field results from large scale internal currents. The field of a permanent magnet results from atomic scale currents.


The field appears only if there is current. It is associated with moving charge.


The Source of the Magnetic Field: Moving Charges

The magnetic field of a charged particle q moving with velocity v is given by the Biot-Savart law:

where r is the distance from the charge and θ is the angle between v and r. (Valid if v<<c.) The Biot-Savart law can be written in terms of the cross product as


The field of an element of circuit

�

dB = µo

4πIds× ˆ r r2

dI=Ids

dB

r

The average B field dB due to the moving charge in an element of circuit of vector length ds carrying current I follows from superposing the fields of the moving charges:


The field of straight wire

11

  All current elements produce B out of page

x

a

�

r = x 2 + a2r

�

dB = µo

4πIds× ˆ r r2

= µo

4πIr2 sinθ

= µo

4πIr2ar

= µo

4πI ax 2 + a2( )3 / 2

�

B = µoIa4π

dxx 2 + a2( )3 / 2

=−∞

∞

∫ µoI2πa

Add them all up:


EXAMPLE 33.4 The magnetic field strength near a heater wire


EXAMPLE 33.4 The magnetic field strength near a heater wire

Twice B(Earth).


Magnetic field lines close upon themselves – they circulate rather them emanate from charges. Although a current loop can appear like a dipole pair of charges, there is no magnetic charge.

Magnetic field lines


Magnetic Dipoles The magnetic dipole moment of a current loop enclosing an area A is defined as

The SI units of the magnetic dipole moment are A m2. The on-axis field of a magnetic dipole is


EXAMPLE 33.7 The field of a magnetic dipole


The superposition of the fields of a stack of loops is a field like that of a bar magnet.

The bar magnetic field results from alignment of many atomic scale electronic currents/magnetic dipoles.

Magnetic materials


Line integrals

Magnetic field lines close upon themselves – they circulate rather them emanate from poles. The line integral of B around a closed loop is a measure of the strength in circulation.


Ampère’s law

Whenever total current Ithrough passes through an area bounded by a closed curve, the line integral of the magnetic field around the curve is given by Ampère’s law:


Ampère’s law example

•  Could have used Ampere’s law to calculate B

r

I

B(r) �

B• ds∫ = Bds =∫ B ds∫ = B2π r= µoI ⇒ B = µoI2πr

Circular path

Surface bounded by path

B||ds B constant on path

path length = 2πr


The strength of the uniform magnetic field inside a solenoid is

where n = N/l is the number of turns per unit length.


Gauss’s law for magnetism

•  Net magnetic flux through any closed surface is always zero

�

B• dA∫ = 0

No magnetic ‘charge’, so right-hand side=0 in the case of magnet field.

�

E • dA∫ = Qenclosed

εo

Compare to Gauss’ law for electric field


General laws of electromagnetism

Gauss’ law Ampere’s law

�

B• dA∫ = 0

�

E• dA∫ = Qenclosed

εo�

B• ds∫ = µoIMagnetostatics

Electrostatics

�

E• ds∫ = ?0

•  Integral of E-field around closed loop is is the change in electric potential going around = zero.

x


The Magnetic Force on a Moving Charge

The magnetic force on a charge q as it moves through a magnetic field B with velocity v is

where α is the angle between v and B.


Motion in a uniform constant magnetic field The magnetic force on a moving charge q is perpendicular to B and to v and results in helical motion. (Circular motion if there is no velocity component along the field.)

Beam of electrons moving in a circle. Lighting is caused by excitation of atoms of gas in a bulb. http://en.wikipedia.org/wiki/Magnetic_field


Derive force by adding forces on charges constituting the current.


Magnetic Forces on Current-Carrying Wires

Consider a segment of wire of length l carrying current I in the direction of the vector l. The wire exists in a constant magnetic field B. The magnetic force on the wire is

where α is the angle between the direction of the current and the magnetic field.


EXAMPLE 33.13 Magnetic Levitation


Interaction between a field and a dipole An current loop in a magnetic field is subject to a net torque aligning the dipole moment with the field.

�

τ = r × F

�

τ = 2

2F sinθ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

F = IB⇒ τ = AIBsinθ

�

A = 2 =loop area

B

F

B

I

�

r

I

F


Interaction between electromagnet and a magnetic substance

An electromagnet can be used to pick up a ferromagnetic material. The field of the electromagnet induces an alignment of the atomic scale dipoles resulting in a net force of attraction.


Does the compass needle rotate clockwise (cw), counterclockwise (ccw) or not at all?

A. Clockwise B. Counterclockwise C. Not at all


A. Clockwise B. Counterclockwise C.  Not at all

Does the compass needle rotate clockwise (cw), counterclockwise (ccw) or not at all?


The magnetic field at the position P points

A. Into the page. B. Up. C. Down. D. Out of the page.


A. Into the page. B. Up. C. Down. D.  Out of the page.

The magnetic field at the position P points


The positive charge is moving straight out of the page. What is the direction of the magnetic field at the position of the dot?

A. Left B. Right C. Down D. Up


A. Left B. Right C.  Down D. Up

The positive charge is moving straight out of the page. What is the direction of the magnetic field at the position of the dot?


What is the current direction in this loop? And which side of the loop is the north pole?

A. Current counterclockwise, north pole on bottom B. Current clockwise; north pole on bottom C. Current counterclockwise, north pole on top D. Current clockwise; north pole on top


A. Current counterclockwise, north pole on bottom B. Current clockwise; north pole on bottom C. Current counterclockwise, north pole on top D. Current clockwise; north pole on top

What is the current direction in this loop? And which side of the loop is the north pole?


A. Left B. Into the page C. Out of the page D. Up E. Down

An electron moves perpendicular to a magnetic field. What is the direction of ?


Which magnet or magnets produced this induced magnetic dipole?

A.  a or d B.  a or c C.  b or d D.  b or c E.  any of a, b, c or d


A.  a or d B.  a or c C.  b or d D.  b or c E.  any of a, b, c or d

Which magnet or magnets produced this induced magnetic dipole?

Chapter 33. The Magnetic Field - Welcome | Department of ... · Magnetostatics ∫B•ds=µ o I...

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