Chapter 30

Sources of the Magnetic Field

Biot-Savart Law

• Biot and Savart arrived at a mathematical expression that gives the magnetic field at some point in space due to a current

• The magnetic field is dB at some point P; the length element is ds; the wire is carrying a steady current of I

Jean-Baptiste Biot1774 – 1862

Félix Savart1791 – 1841

Biot-Savart Law

• Vector dB is perpendicular to both ds and to the unit vector directed from ds toward P

• The magnitude of dB is inversely proportional to r2, where r is the distance from ds to P

• The magnitude of dB is proportional to the current and to the magnitude ds of the length element

r̂

Biot-Savart Law

• The magnitude of dB is proportional to sin where is the angle between the vectors ds and

• The observations are summarized in the mathematical equation called the Biot-Savart law (magnetic field due to the current-carrying conductor):

• µo = 4 x 10-7 T.m / A: permeability of free space

r̂

24oμ d

dπ r

s rB

ˆI

Biot-Savart Law

• To find the total field, sum up the contributions from all the current elements

24oμ d

dπ r

s rB

ˆI

24oμ d

π r

s rB

ˆI

Biot-Savart Law• The magnitude of the magnetic field varies as the

inverse square of the distance from the ds element

• The electric field due to a point charge also varies as the inverse square of the distance from the charge

• The electric field created by a point charge is radial in direction

• The magnetic field created by a current element is perpendicular to both the length element and the unit vector

• The current element producing a magnetic field is part of an extended current distribution

A Long, Straight Conductor• The thin, straight wire is carrying a constant current

kdxrsd ˆ)cos(ˆ

kr

dxIBd ˆcos

4 20

cos

ar

tanax

2cos

addx

dk

a

Icosˆ

40

2

1

cos40

d

a

IB

A Long, Straight Conductor• The thin, straight wire is carrying a constant current

• If the conductor is an infinitelylong, straight wire, θ1 = π/2 and

θ2 = – π/2 , and the field becomes

2

1

cos40

d

a

IB

210 sinsin4

a

I

a

IB

20

2

1

cos40

d

a

IB

A Long, Straight Conductor

• The magnetic field lines are circles concentric with the wire

• The field lines lie in planes perpendicular to the wire

• The magnitude of the field is constant on any circle of radius a

• Right Hand Rule #2: Grasp the wire in your right hand and point your thumb in the direction of the current and your fingers will curl in the direction of the field

A Curved Wire Segment

• Find the field at point O due to the wire segment (I, a are constants)

• The field at the center of the full circle loop

20

4 a

IdsdB

dsa

IB

20

4

sa

I2

0

4

aa

I2

0

4

a

I

40

2

40

a

IB

a

IB

20

Magnetic Field of a Current Loop

Magnetic Force Between Two Parallel Conductors

a

IB

2

202

lIBF 121 lIa

I1

20

2

a

II

l

F

2

2101

Magnetic Force Between Two Parallel Conductors

• The force (per unit length) on wire 1 due to the current in wire 1 and the magnetic field produced by wire 2:

• Parallel conductors carrying currents in the same direction attract each other

• Parallel conductors carrying currents in the opposite directions repel each other

a

II

l

FB

2

210

Chapter 30Problem 19

Two long, parallel wires are attracted to each other by a force per unit length of 320 μN/m when they are separated by a vertical distance of 0.500 m. The current in the upper wire is 20.0 A to the right. Determine the location of the line in the plane of the two wires along which the total magnetic field is zero.

Ampere and Coulomb revisited

• The force between parallel conductors can be used to define the Ampere (A): If two long, parallel wires 1 m apart carry the same current, and the magnitude of the magnetic force per unit length is 2 x 10-7 N/m, then the current is defined to be 1 A

• The SI unit of charge, the Coulomb (C), can be defined in terms of the Ampere: If a conductor carries a steady current of 1 A, then the quantity of charge that flows through any cross section in 1 second is 1 C

Ampère’s Law• Ampère’s Circuital Law: a procedure

for deriving the relationship between the current in an arbitrarily shaped wire and the magnetic field produced by the wire

• Choose an arbitrary closed path around the current and sum all the products of B|| Δℓ around the closed path (put the thumb of your right hand in the direction of the current through the loop and your fingers curl in the direction you should integrate around the loop)

IsdB 0

Ampère’s Law for a Long Straight Wire

• Use a closed circular path

• The circumference of the circle is 2 r

dsBsdB

I0rB 2

r

IB

20

Ampère’s Law for a Long Straight Wire

'0IrB 2

rR

IB

20

2

2

2'

R

r

I

I

IR

rI

2

2

'

IR

r2

2

0 sdB

Chapter 30Problem 24

A long, straight wire lies on a horizontal table and carries a current of 1.20 μA. In a vacuum, a proton moves parallel to the wire (opposite the direction of the current) with a constant velocity of 2.30 × 104 m/s at a constant distance d above the wire. Determine the value of d.

(You may ignore the magnetic field due to Earth.)

Magnetic Field of a Solenoid• If a long straight wire is bent into a coil of

several closely spaced loops, the resulting device is called a solenoid

• It is also known as an electromagnet since it acts like a magnet only when it carries a current

• The field inside the solenoid is nearly uniform and strong – the field lines are nearly parallel, uniformly spaced, and close together

• The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid

Magnetic Field of a Solenoid• The field lines of the solenoid resemble

those of a bar magnet

• The magnitude of the field inside a solenoid is approximately constant at all points far from its ends

B = µo n I

• n = N / ℓ : the number of turns per unit length

• This result can be obtained by applying Ampère’s Law to the solenoid

Magnetic Field of a Solenoid• A cross-sectional view of a tightly wound

solenoid

• If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero

• Apply Ampère’s Law to the blue dashed rectangle

1path

sdBsdB

1path

dsB

Il

NB 0 nI0

BlNI0

Magnetic Flux• Magnetic flux associated with a magnetic field is

defined in a way similar to electric flux

• SI unit of flux: Weber

• Wb = T. m²

AdBB

Wilhelm Eduard Weber1804 – 1891

• For a flat surface with an area A in a uniform magnetic field, the flux is (θ is the angle between B and the normal to the plane):

ΦB = BA = B A cos θ

• When the field is perpendicular to the plane, θ = 0 and ΦB = ΦB, max = BA

• When the field is parallel to the plane, θ = 90° and ΦB = 0

• The flux can be negative, for example if θ = 180°

Magnetic Flux

• The value of the magnetic flux is proportional to the total number of magnetic field lines passing through area

• When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum

• When the area is parallel to the lines, no lines pass through the area and the flux is 0

Magnetic Flux

• Magnetic fields do not begin or end at any point

• The number of lines entering a surface equals the number of lines leaving the surface

• Gauss’ law in magnetism says the magnetic flux through any closed surface is always zero:

Gauss’ Law in Magnetism

0 AdBB

Magnetic Effects of Electrons – Orbits • An individual atom should act like a magnet because of

the motion of the electrons about the nucleus

• Each electron circles the atom once in about every 10-16 seconds; this would produce a current of 1.6 mA and a magnetic field of about 20 T at the center of the circular path

• However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom

• The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

Magnetic Effects of Electrons – Spins • Electrons also have spin (it is a

quantum effect)

• The classical model is to consider the electrons to spin like tops

• The field due to the spinning is generally stronger than the field due to the orbital motion

• Electrons usually pair up with their spins opposite each other, so their fields cancel each other, hence most materials are not naturally magnetic

Magnetic Effects of Electrons – Domains • In some materials – ferromagnetic – the spins do not

naturally cancel

• Large groups of atoms in which the spins are aligned are called domains

• When an external field is applied, it causes the material to become magnetized: the domains that are aligned with the field tend to grow at the expense of the others

Domains and Permanent Magnets• In hard magnetic materials, the domains remain

aligned after the external field is removed

• The result is a permanent magnet

• In soft magnetic materials, once the external field is removed, thermal agitation causes the materials to quickly return to an unmagnetized state

• With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field

Ferromagnetism

• Some substances exhibit strong magnetic effects called ferromagnetism (e.g., iron, cobalt, nickel, gadolinium, dysprosium)

• They contain permanent atomic magnetic moments that tend to align parallel to each other even in a weak external magnetic field

Paramagnetism

• Paramagnetic substances have small but positive magnetism, which results from the presence of atoms that have permanent magnetic moments

• These moments interact weakly with each other

• When placed in an external magnetic field, atomic moments tend to line up with the field and the alignment process competes with thermal motion which randomizes the moment orientations

Diamagnetism

• When an external magnetic field is applied to a diamagnetic substance, a weak magnetic moment is induced in the direction opposite the applied field

• Diamagnetic substances are weakly repelled by a magnet

Earth’s Magnetic Field

• The Earth’s geographic north (south) pole corresponds to a magnetic south (north) pole – a north (south) pole should be a “north- (south-) seeking” pole

• The Earth’s magnetic field resembles that achieved by burying a huge bar magnet deep in the Earth’s interior

• The most likely source of the Earth’s magnetic field – electric currents in the liquid part of the core

Earth’s Magnetic Field

• The magnetic and geographic poles are not in the same exact location – magnetic declination is the difference between true north (geographic north pole) and magnetic north pole

• The amount of declination varies by location on the earth’s surface

• The direction of the Earth’s magnetic field reverses every few million years (the origin of these reversals is not understood)

Answers to Even Numbered Problems

Chapter 30:

Problem 2

200 nT

Answers to Even Numbered Problems

Chapter 30:

Problem 18

(a) 10.0 µT out of the page (b) 80.0 µT toward wire 1(c) 16.0 µT into the page(d) 80.0 µT toward wire 2

Answers to Even Numbered Problems

Chapter 30:

Problem 54

1.80 mT