Prerequisites for Magnetism Before you start to study - - PowerPoint PPT Presentation

Prerequisites for Magnetism

• Before you start to study electromagnetism you must

review these topics:

• Cross product of two vectors: modulus and direction

(right-hand rule and screw rule).

• Torque of a vector (force).
• Surface vector of a flat surface.

Unit 6: Magnetic forces

• Introduction. Magnetic Field.
• Forces exerted by a magnetic field:
• Force on a moving electric charge.
• Application: Hall effect.
• Force on a conductor flowed by a current.
• Effect of a uniform magnetic field on a flat current-

carrying loop. Magnetic moment. Electric motor.

• Magnets: Poles attract or repel each other in the

same way as electric charges.

• Poles of a magnet: north pole and south pole

(similar to + and – charges).

• Introduction. Magnetic field

Differences Magnetic – Electric poles:

Tipler, chapter 26

• A

magnetic pole can’t be isolated. Magnetic monopoles don’t exist.

• If you break a magnet in two pieces, both pieces

have their north and south poles. Where the effects of magnetic poles appear, this is said to be a Magnetic Field (B). A magnetic field can be represented as a vector B at each point in the

• field. Field lines are parallel to B, as in an electric

field.

• When an electric charge q is moving with velocity v, inside a

magnetic field B, a force F acts on the charge, given by (empirical result) :

Tipler, chapter 26, section 26.1

Force on a moving charge in a magnetic field

B

v

F

α

+

q

Right-hand rule

• r

screw rule

B v F

• ×

= q

B

v

F

α

• q

If v and B are parallel F=0

[ ]

1 2

] [ ] [ ] [

− −

= = I MT q v F B

• Unit I.S: Tesla.
• Other unit used: Gauss

1 G=10-4 T.

• Earth field: 0,5× 10-4 T .
• Magnetic resonance: 2~5 T.

So, a magnetic force ( ) acts on charge carriers. Let’s suppose such carriers are electrons (-).

J vd Fm B

It occurs when a current is flowing through a conductor inside a magnetic field perpendicular to I.

Application: Hall effect

B v F

• ×

=

d m

q

m

F

• This magnetic force separates positive and negative charges

perpendicularly to the electric current and magnetic field.

I

Vd is drift speed

B qv q

d d m

= = B v F

• Tipler, chapter 26, section 26.4

VH J vd Fm B

When charges are split up, an electric field appears (and so a drop in potential), producing an electric force ( ) opposite to the magnetic force:

Application: Hall effect

E F

• q

e =

e

F

• Fe

E

I

When equilibrium is reached, magnetic and electric forces are balanced: Drop in potential VH is Hall voltage (order of μV):

B v E qE B qv F F

d d e m

=  =  = nqb IB Ba nqS I Ba d v Ea H V = = = =

a b

VH J va F B

⊕ ⊕ ⊕ ⊕

• VH

J va F B

Negative charge carriers

The type of charge carriers determine the polarity of VH (useful to distinguish n or p semiconductor type).

Application: Hall effect

Positive charge carriers

I I

It is also useful to build speed selectors. Only those charges with that speed cancelling electric and magnetic forces will cross the conductor without deflection.

• When a conductor with current flowing through it is inside a magnetic

field, the force on the conductor is the sum of the forces acting on all the charges in the conductor:

Force on a conductor with current inside a B

Tipler, chapter 26, section 26.1

• Taking an infinitesimal piece of conductor (lenght dl), the total number
• f charge carriers in dl will be:

n= density of charge carriers

dl S n

And the force acting over dl:

( )

B l B v F

• ×

= × = d I dl S n q d

d

S nqv JS I

d

= =

BE CAREFUL: dl is a vector in the same direction than vd and I

is called a current element

l

• Id

B

• If we consider a conductor with any shape, the total force
• n

the conductor will be the sum

• f

all the forces (integration) on it:

 

× = × =

b a b a

d I d I B l B l F

• )

(

l d

a b

Force on a conductor with current

B

• Let’s take a straight conductor inside a uniform magnetic

field B:

B l B l B l F

• ×

= × = × =

 

I d I d I

b a b a

) (

l

• d

a b

l

• I

l is a vector from a to b, in the same direction than the current.

F

• Example: Force on a straight conductor in a uniform B

I and B are constants from a to b

a b

• B

I

Effect of a uniform magnetic field on a flat current- carrying loop. Magnetic moment.

Tipler, chapter 26, section 26.3

• Let’s take a flat loop inside a uniform B, with a

current I:

B b F

• ×

= I

1 1

F

• 2

F

• B

3

F

• 4

F

• a
• b
• 1

2

F B b F

• −

= × − = ) ( I

3 4

F B a F

• −

= × = I B a F

• ×

− = ) ( I

3

• On each side of the loop, a force will act:
• Resulting force on loop is zero:

4 1

= = 

= i i

F F

• Effect of a uniform magnetic field on a flat current-

carrying loop. Magnetic moment.

1 1

2 F a τ

• ×

− =

1

F

• 2

F

• B

3

F

• 4

F

• a
• b
• Torque of each force in relation to point O (loop centre):

O

3 3

2 F b τ

• ×

− =

2 2

2 F a τ

• ×

=

4 4

2 F b τ

• ×

=

Effect of a uniform magnetic field on a flat current- carrying loop. Magnetic moment.

magnetic moment of the loop surface vector of the loop (direction given by

current in loop according to right-hand rule)

B a b τ

• ×

× = = 

=

) ( I

i i 4 1

τ

• Resulting torque about O is:
• So, the resulting torque in the loop can be written as:

S μ

• I

= a b S

• ×

= B μ τ

• ×

=

[ ]

2

IL = µ

Effect of a uniform magnetic field on a flat current- carrying loop. Magnetic moment.

S

• I

S μ

• I

=

I

S

• S

μ

• I

=

• If there isn’t just one loop, but many (N) loops (a coil):

S μ

• NI

=

• So, when we apply an I to a flat loop in a uniform B, the

loop turns until magnetic moment µ is aligned with B (τ=0), the loop remaining in equilibrium.

Effect of a uniform magnetic field on a flat current- carrying loop. Magnetic moment.

• B
• B

An electric motor consists of a set of loops carrying a current inside a magnetic field. Its behaviour is that of a receptor. Features: ε’ and r’

Application: electric engine B S iN B

• ∧

= ∧ = µ τ

ε’ and ω are directly related

ω ε ε ω ε ω τ ' k ' i ' kiNSB i ' P

m

=  =  = =