1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole...

21
1 Chapter 17 Chapter 17 Magnetic Field Magnetic Field and and Magnetic Forces Magnetic Forces

Transcript of 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole...

Page 1: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

1

Chapter 17Chapter 17

Magnetic FieldMagnetic Field

and and

Magnetic ForcesMagnetic Forces

Page 2: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

2

Magnetism

S

N

South

NorthSouth magnetic pole

South geographic pole

Earth’s magnetic field

North magnetic pole

North geographic pole

Opposite poles : attract each other

Like poles: repel each other

Northnorth pole

south pole

The needle of a compass aligns with the magnetic field

Earth is a magnetic. The axis of earth’s magnetic is not parallel to

its geographic axis

Magnetic declination

(Rotation axis)

(a vector field)

Page 3: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

3

N

S

N

S

F

F

Attract each other

S

N

N

S

F

F

Repel each other

S

N

S

N

F

F

Attract each other

N

S

S

N

F

F

Repel each other

Page 4: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

4

Magnetic Field

In addition to the electric field, a moving charge or a current in space can create a magnetic field.

An electric force (F = Q0E) will exert on other charge (Q) present in the electric field (E). Similarly, the magnetic field also exerts a magnetic force on other moving charge or current present in the magnetic field.

Oersted’s Experiment

N

S

I = 0 (no current)

EW

I 0I

N

S

EW

I 0

I

N

S

EW

Page 5: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

5

Magnitude of magnetic force is proportional to:

• magnitude of the charge• magnitude or strength of the magnetic field• velocity of the moving particle (for electric force, it is the same no

matter the charge is moving or not) or the component of velocity perpendicular to the field.

A charged particle at rest will have no magnetic force.

The direction of magnetic force (F) is not the same as the direction of magnetic field (B). Instead, the magnetic force is always perpendicular to both direction of magnetic field (B) and the velocity (v).

Direction of B: the north pole direction of a compass needle.

For a magnet, the direction of B is pointing out of its north pole and into its south pole.

S N

Page 6: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

6

BvQF

QBvBvQF

sin

Magnetic force on a moving charged particle:

where:

F : magnetic force [N]Q : magnitude of charge [C]v : velocity of the charge [m/s]B : magnetic field [T or Ns/Cm or N/Am (A: ampere)]

1 N/Am = 1 tesla = 1 T [Nikola Tesla (1857 – 1943)]

Right-hand rule

Magnetic field (B)+Q

Force (F)

Velocity (v)

v

Positive charge

Right hand rule

cross product

Page 7: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

7

Magnetic field (B)_-Q

Force (F)

Velocity (v)

v

negative charge

For a negative charge, the force is opposite to the case of the positive charge.

sinQBv

BvQF

Page 8: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

8

Magnetic Flux

0 AdBA

Total magnetic flux through a surface A:

AdBdABdABA

cos Sum of magnetic flux thru

areas of all elements

B

B

dA

||B

Where:

A: magnetic flux (a scalar) [weber (Wb)]

B : magnetic field [T]A : surface area [m2]

1 Wb = 1 T m2 = 1 Nm / A

Total magnetic flux through a closed surface = 0

Gauss’s Law for Magnetism

dA

dB A Magnetic field B is also

called magnetic flux density

phi"":

In = Out

Page 9: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

9

Motion of Charged Particles in a Magnetic Field

A charge particle under the action of a magnetic field only moves with a constant speed. The motion is determined by Newton’s laws of motion.

Circular motion of a positive charge in a uniform magnetic field (B):

“x” denotes that the magnetic field is pointing into the plane

v1

v2

P1

P2

R

R

0

s

R

varad

2

BQ

mvR

R

vmmavBQF

2

m : mass of the particlev : constant velocity R : radius of the circular orbit

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

F

v

+F

v+

R

Page 10: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

10

Magnetic Force on a Conductor

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

F

v

+

L

Q

A

Conductor with current

I

Magnetic force on a straight wire:

sinILBILB

BLIF

Where:

F : magnetic forceI : total currentL : length of the wire segment

Magnetic force on an infinitesimal wire (not straight wire):

BLIdFd

Divide the wire into infinitesimal straight line

IL

cos|| BB

sinBB

F

B

Page 11: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

SOURCES OF MAGNETIC FIELD

11

“Source point” is referred to the location of a charge (Q) moving with a constant velocity (v) in a magnetic field.

“Field point” is referred to the location or point where the field is to be determined, e.g. location of point “k”.

Magnetic field of a point charge moving with a constant velocity:

20

20 sin

4

ˆ

4 r

Qv

r

rvQB

+

B

v

k

r

Q

BB = 0

where:B : magnetic field Q : point chargev : velocity of the charger : distance from the charge to the field

point

runit vecto : r̂0 = 4 10-7 Ns2/C2

1 Ns2/C2 = 1 Wb/Am = 1 Tm/A = 1 n/A2

xI

B

+ Q

charge is moving into the plane

Page 12: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

12

Applying the principle of superposition, the magnetic fields of a number of moving charges can be calculated.

Total magnetic field due to a number of moving charges = vector sum of the electric fields due to the individual charges

B

k

r

dBdB = 0

dL

20

20

20

ˆ

4

sin

4

ˆ

4

r

rLIdB

r

IdL

r

rLIddB

Law of Biot and Savart for Magnetic Field of a Current Element (B):

wheredL : represents a short segment of a current-

carrying conductorI : current in the segment

runit vecto : r̂AnQvI d nQ : total charges

vd : drifting velocityA : cross-section area of segment

Page 13: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

13

Magnetic field of a straight current-carrying conductor:

conductor) carrying-currentstraight a(for 2

then x, L , If

2

4

)(4

)sin(sin

;

ˆ

4

sin

4

ˆ

4

0

22

22

0

2/3220

22

22

20

20

20

x

IB

LLxL

Lxx

LIB

dyyx

xIB

yx

x

dydLyxr

r

rLIdB

r

IdL

r

rLIddB

L

L

x

y

0x

-L

+L

dL 22 yxr

dB

I

r

- y

Page 14: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

14

Example 17.1:The figure shows an end view of two parallel wires carrying the same current I in opposite directions. Determine the magnitude and direction of magnetic flux B at point A.

Solution:Use principle of superposition of magnetic fields:

Btotal = B1 + B2

Point A is closer to wire 1 than to wire 2, the field magnitude B1 > B2

L

I

L

IB

L

I

L

IB

8)4(2

4)2(2

002

001

Use right hand rule, B1 is in the – y-direction and B2 is in the + y-direction. As B1 > B2, Btotal is in the – y-direction and the magnitude is:

L

I

L

I

L

IBBBtotal

884000

21

x

2L 2L

Wire 1 Wire 2I I

B1

B2

A

y

xI I

B

x : into the plane : out of the plane

Page 15: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

15

Ampere’ Law

The line integral of magnetic field intensity around a single closed path is equal to the algebraic sum of currents enclosed.

enclILdB 0

r1

r21 23 4

B

Integration path not enclosing the conductor

rI

B

B

dL

B

B

dL

I

rr

I

dLB

dLBLdB

0

0

||

)2(2

Integration path enclosing the conductor

Page 16: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

19

Magnetic Field of a Circular Current Loop

Page 17: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

20

The Solenoid

A long wire wound in the form of a helical coil is known as a solenoid.

I

I

P

R

θ 2θ

θ 1

I

y

dy

(b)(a)

Page 18: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

1

(y2+R2) 1/2

y2

(y2+R2) 3/2

R2

(y2+R2) 3/2

Also, the magnetic field dB due to the current dI in dy can be found as,

From the figure, the current for a length increment dy is

dyL

NIdI

dIRy

RdB

2322

20

)(2

2322

20

)(2 Ry

dy

L

NIRdB

ddyRy

R

d

dy

Ry

y

Ryd

dy

dy

d

Ry

y

cos)(

cos)()(

1sin

)(sin

2322

2

2322

2

2122

2122

Page 19: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

22

So, dB = ( ) cos θ dθ μ0 I N

2 L

=> B = ( ) ∫ cos θ dθ μ0 I N

2 L

θ2

θ1

= ( ) (sin θ2 – sin θ1 ) μ0 I N

2 L

B = (sin θ2 – sin θ1 ) jμ0 n I

2

where n = , number of turns per unit lengthN

LThis formula represents the magnetic field along the centroid axis of a finite solenoid.

For infinite long solenoid, it is assumed that θ1 = -π/2 and θ2 = π/2.

B = μ0 n I j

Page 20: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

23

Example 17.3 :

Since the length of the solenoid is quite large in comparative with its diameter, the magnetic field near its middle is approximately uniform. It is therefore reasonable to consider it as a case of infinite solenoid

B = μ0 n I j.

Solution

A solenoid has 300 turns wound around a cylinder of diameter 1.20 cm and length 14.0 cm. If the current through the coils is 0.410 A, what is the magnitude of the magnetic field inside and near the middle of the solenoid.

The number of turns per unit length (n) is

n = N/L = (300 turns) / (0.14 m)

= 2.14 × 103 turns / m

Therefore, the magnetic field inside and near the middle of the solenoid is,

B = μ0 n I j = (4π × 10-7 Tm/A) (2.14 × 103 turns / m) (0.410 A) = 1.10 × 10-3 T

Page 21: 1 Chapter 17 Magnetic Field and Magnetic Forces. 2 Magnetism S N South North South magnetic pole South geographic pole Earth’s magnetic field North magnetic.

24

I

I

N

S

I

I

S

N

Magnetic fields of a finite solenoid