Typical magnetic field strengths - SMUdclarke/PHYS1101/documents/magnetism_ppt.pdf · Typical magnetic field strengths source of magnetism surface of a neutron star a superconducting

PHYS 1101, Winter 2009, Prof. Clarke 1

Typical magnetic field strengths

source of magnetism

surface of a neutron star

a superconducting magnet

a laboratory electromagnet

a refrigerator magnet

surface of the Earth

interstellar space

in a magnetically shielded room

Tesla

108

10

0.1 - 1.5

5 × 10–3

5 × 10–4

10–10

10–14


magnetic dipole

vd

E i

FB

w d

B

vd

E i

FB

w

d

B B

the Hall effect

Magnetic dipole field lines and the Hall effect


v FB B

x

y

z

v FB m, q

positive charge orbits clockwise within a positive B

negative charge orbits counter-clockwise within a positive B

a length of wire, L, carrying a current, i, is embedded in an external magnetic field, B.

Larmor motion and force on a wire

B

L

x

x0

vd , i

B


a

b

1

2

3

4

i

τ, α, ω

x

y

z

B FB 3

FB 1

i

i i

FB = FB = 0 (L || B) 2 4

N S κ

coil

B

Torque on a square current loop immersed in a uniform magnetic field

schematic diagram of a galvanometer

Current loop and galvanometer


i

ds r P θ

dB

magnetic field from a general piece of wire

magnetic field at the centre of curvature of a circular arc of wire

ds

P

r

i

dB φ

dφ

magnetic field around a long, straight wire

The Law of Biot and Savart

ds

P dB

r

i s

θ φ

R

dB


x

y

z

d

a

b

ia

ib

Fab

Fba

B from wire b

B from wire a

d

a

b

ia

ib

Fab

Fba

B from wire b B from wire a x

y

z

parallel wires

antiparallel wires

Force on (anti)parallel wires


a b

i i

x y

z

2d 2d Bb Ba

y

x z

a b d –d

By

x a b d –d

i i

Combined magnetic field from antiparallel currents


Clicker question 1

i

P

The magnetic field at point P points:

a) up b) down c) left

d) right e) into the page f) out of the page


Clicker question 1

i

P





P




Clicker question 2

A positive charge moves straight out of the page. v


P




Clicker question 2

A positive charge moves straight out of the page. v


a

b

1

2

3

4

i

x

y

z

B i

i i

Magnetic force on a wire: FB = i L × B

where L is a vector with length of the wire and pointing in the direction of the current.

Clicker question 3

a) Force on wire 1 is iaB and points in the –y-direction b) Force on wire 2 is ibB and points in the +x-direction c) Force on wire 3 is iaB and points in the –z-direction d) Force on wire 3 is iaB and points in the –y-direction e) None are true. f) All are true

Considering only the force exerted on the wires by the background magnetic field, B, which of the following statements is true?


a

b

1

2

3

4

i

x

y

z

B i

i i

Magnetic force on a wire: FB = i L × B

where L is a vector with length of the wire and pointing in the direction of the current.

Clicker question 3

a) Force on wire 1 is iaB and points in the –y-direction b) Force on wire 2 is ibB and points in the +x-direction c) Force on wire 3 is iaB and points in the –z-direction d) Force on wire 3 is iaB and points in the –y-direction e) None are true. f) All are true

Considering only the force exerted on the wires by the background magnetic field, B, which of the following statements is true?


Clicker question 4

Which current segment, if any, produces no magnetic field at point P?

d) None of the current segments produce a magnetic field at P.

e) All current segments produce a magnetic field at P.

Law of Biot and Savart: dB = i ds × r r2

µ0 4π

i P b) ds r

i P c)

ds r

P

r

i a)

ds


Clicker question 4

Which current segment, if any, produces no magnetic field at point P?

d) None of the current segments produce a magnetic field at P.

e) All current segments produce a magnetic field at P.

Law of Biot and Savart: dB = i ds × r r2

µ0 4π

i P b) ds r

i P c)

ds r

P

r

i a)

ds


ds

iencl

B

ampèrian loop: can be any closed loop around current

Ampère’s law

Direction around loop is given by the right-hand-rule: Place the right thumb in the direction of iencl, fingers then curl in the direction to traverse loop.

As with Gauss’ Law for E, Ampère’s Law can be used to calculate B only for highly symmetric problems.

B ⋅ ds = µ0 iencl ∫


Magnetic field around and within a long straight wire

B

B

r

i

B

ampèrian loop B x

y

z

r

R

J

ampèrian loop

J

A i

definition of current density

i = J ⋅ dA ∫ A


i i

The solenoid

∞ ∞ d

c

b a

e

f

ampèrian loop

L


The solenoid as a magnetic dipole


i

i r

ampèrian loop

i

B

B

The toroid

pictorial representation

top view, cut-away


iind

Bind v

iind

Bind v

Lenz’ Law

When magnetic flux through a loop increases, induced current in the loop is such that induced magnetic field opposes the increasing flux.

When magnetic flux through a loop decreases, induced current in the loop is such that induced magnetic field adds to the decreasing flux.

Induced flux works to minimise the rate of change of flux through the loop.


B

ω

B

Examples of time-varying magnetic fluxes

A loop spinning at a constant angular speed, ω,

in a uniform B sets up a sinusoidally varying ΦB

A rail sliding at a constant speed, v, in a uniform B sets up a linearly

varying ΦB

A time-varying i creates a

time-varying B and thus ΦB

~

i

i

i

i

B(t)

v


B

ω

A θ

Clicker question 5

At the moment shown in the figure, the loop is spinning so that the angle, θ, between the area vector of the loop, A, and the background magnetic field, B is decreasing with time. Which of the following statements is true?

Bind

i

a) The induced current in the loop is clockwise and the induced magnetic field is parallel to A.

b) The induced current in the loop is counterclockwise and the induced magnetic field is parallel to A.

c) The induced current in the loop is clockwise and the induced magnetic field is antiparallel to A.

d) The induced current in the loop is counterclockwise and the induced magnetic field is antiparallel to A.

Bind

i

Bind

i

Bind

i


Clicker question 5

At the moment shown in the figure, the loop is spinning so that the angle, θ, between the area vector of the loop, A, and the background magnetic field, B is decreasing with time. Which of the following statements is true?

Bind

i

a) The induced current in the loop is clockwise and the induced magnetic field is parallel to A.

b) The induced current in the loop is counterclockwise and the induced magnetic field is parallel to A.

c) The induced current in the loop is clockwise and the induced magnetic field is antiparallel to A.

d) The induced current in the loop is counterclockwise and the induced magnetic field is antiparallel to A.

Bind

i

Bind

i

Bind

i

i

B

ω

A θ

Bind

As θ decreases, ΦB increases so Bind is set up to offset that increase.

PHYS 1101, Winter 2009, Prof. Clarke

Bind i

25

Clicker question 6

At the moment shown in the figure, the rail is sliding to the right so that the area of the loop is decreasing with time. The rail and loop are conductors and electrically connected. Which of the following statements is true?

a) The induced current in the loop is clockwise and the induced magnetic field is parallel to B.

b) The induced current in the loop is counterclockwise and the induced magnetic field is parallel to B.

c) The induced current in the loop is clockwise and the induced magnetic field is antiparallel to B.

d) The induced current in the loop is counterclockwise and the induced magnetic field is antiparallel to B.

Bind i

Bind i

Bind i

B v


Bind i

26

Clicker question 6

At the moment shown in the figure, the rail is sliding to the right so that the area of the loop is decreasing with time. The rail and loop are conductors and electrically connected. Which of the following statements is true?

a) The induced current in the loop is clockwise and the induced magnetic field is parallel to B.

b) The induced current in the loop is counterclockwise and the induced magnetic field is parallel to B.

c) The induced current in the loop is clockwise and the induced magnetic field is antiparallel to B.

d) The induced current in the loop is counterclockwise and the induced magnetic field is antiparallel to B.

Bind i

Bind i

Bind i

B Bind

i

i i

i

Flux ΦB decreases as the area of the loop decreases, so Bind is set up to offset that decrease.

v


iind

iind

Bind

iind

iind

Bind

27

iind

iind

Bind

iind

iind

Bind

i decreasing

i decreasing

i increasing

A

B

C

D

1

3

4

Clicker question 7

Match ‘em up!! Which of the loops, A, B, C, D, should replace the grey loops in 1, 2, 3, 4?

a) 1-C, 2-A, 3-B, 4-D

b) 1-A, 2-C, 3-D, 4-B

c) 1-B, 2-B, 3-C, 4-C

d) 1-D, 2-A, 3-A, 4-D

e) 1-C, 2-C, 3-B, 4-B

f) 1-B, 2-C, 3-C, 4-B

i increasing ~

i

i

i

i

2

B(t)

i

i

~

i

i

B(t)

i

i

~

i

i

B(t)

i

~

i i

i

B(t)


iind

iind

Bind

iind

iind

Bind

i decreasing

i decreasing

i increasing

A

B

C

D

1

3

4

Clicker question 7

i increasing

2

✗ ~

i

i

i

i

iind

Bind B(t)

i

i

~

i

i

iind

Bind B(t)

i

i

i

~

i

iind

Bind B(t)

i

~

i i

i

iind

Bind B(t)

Match ‘em up!! Which of the loops, A, B, C, D, should replace the grey loops in 1, 2, 3, 4?

a) 1-C, 2-A, 3-B, 4-D

b) 1-A, 2-C, 3-D, 4-B

c) 1-B, 2-B, 3-C, 4-C

d) 1-D, 2-A, 3-A, 4-D

e) 1-C, 2-C, 3-B, 4-B

f) 1-B, 2-C, 3-C, 4-B

iind

iind

Bind

iind

iind

Bind ✗


v

29

Clicker question 8

A square conducting loop is dragged at a constant velocity, v, along a path taking it through a region of uniform magnetic field, B, as shown. Considering the four “snapshots” shown in the figure, which statement is correct?

a) F2 = F4 > F3 = F1 b) F1 = F2 = F3 = F4

c) F3 > F4 > F2 > F1 d) F3 > F4 = F2 > F1

v v

B

v F1 F4 F2 F3


Clicker question 8

A square conducting loop is dragged at a constant velocity, v, along a path taking it through a region of uniform magnetic field, B, as shown. Considering the four “snapshots” shown in the figure, which statement is correct?

a) F2 = F4 > F3 = F1 b) F1 = F2 = F3 = F4

c) F3 > F4 > F2 > F1 d) F3 > F4 = F2 > F1

F1 v

F4 F2 v

F3 v

B iind iind

Bind Bind

v


Clicker question 9

A conducting loop is incompletely embedded in a region of uniform magnetic field, B. If B begins to increase rapidly in strength, what happens to the loop? a) The loop is pulled to the right, deeper into the magnetised region. b) The loop is pushed to the left and expelled from the magnetised region. c) The tension in the sides of the loop increase, but the loop doesn’t move. d) The loop is pushed upward. e) The loop is pushed downward. f) Nothing.

B

Bind iind F


Clicker question 9

A conducting loop is incompletely embedded in a region of uniform magnetic field, B. If B begins to increase rapidly in strength, what happens to the loop? a) The loop is pulled to the right, deeper into the magnetised region. b) The loop is pushed to the left and expelled from the magnetised region. c) The tension in the sides of the loop increase, but the loop doesn’t move. d) The loop is pushed upward. e) The loop is pushed downward. f) Nothing.

B

Bind iind

Note that expelling the loop helps to counter the increasing flux, which is consistent with Lenz’ Law.

F


R

+

– ε

a

b

i

i

Estat

Estat

Estat

Eε R

a

b

iind

iind

Eind

Eind

Eind

Eind Bind

dΦB dt

A simple i-R circuit with a seat of emf, ε, that sets up a static electric field, Estat

Estat is conservative

A simple i-R circuit embedded in a time-varying magnetic flux, ΦB, that induces an electric field, Eind

Eind is not conservative

Static and induced electric fields


Faraday’s law

Eind

B

ds

C

A

The time-rate-of-change of the magnetic flux, ΦB, passing through

an arbitrary surface of area A is equal to

the induced emf, εind, about the circumference, C, of that loop

which in turn is equal to the closed line integral of the

induced electric field, Eind, about C.

= B ⋅ dA = – εind = – Eind ⋅ ds dΦB dt

d dt

Lenz’ Law

∫ C ∫ A


Applications of Faraday’s law

L

iind

F

Bind

v

F1

F2

F3

Bext

x

a

R A0

Bext

f

A rectangular loop is pulled from a region with a uniform magnetic field, Bext

A semi-circular portion of the circuit is rotated within a uniform magnetic field, Bext


Maxwell’s Equations (in integral form)

“To anyone who is motivated by anything beyond the most narrowly practical, it is worth while to understand Maxwell’s equations simply for the good of the soul.”

J. R. Pierce, 1956, in Electrons, Waves, and Messages (Hanover House)

Lenz’ Law

Gauss’ Law for electric field

Gauss’ Law for magnetic field

Faraday’s Law

Ampère-Maxwell Law

the “displacement current”, id

B ⋅ ds = µ0 iencl + µ0ε0 dΦE dt

∫ C

E ⋅ ds = – dΦB dt

∫ C

B ⋅ dA = 0 ∫ A

E ⋅ dA = qencl ε0

∫ A

Documents

Typical magnetic field strengths - SMUdclarke/PHYS1101/documents/magnetism_ppt.pdf · Typical magnetic field strengths source of magnetism surface of a neutron star a superconducting