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AAST/AEDT

AP PHYSICS B:

MAGNETIC FIELD

Let us run an experiment. We place two parallel wires close to each other. If we turn the

current on, the wires start to interact. If currents are opposite by their direction then the

repulsion is observed. Currents attract if the directions are equal. If the current exists only

in one wire, then there is no interaction. We also have no interaction if the current

interacts with another current in twisted wires.

The conclusion from those observations is:

Current changes the properties of the surrounding space. That change creates a

force that is exerted on any other current placed into this space. The space with the

property to exert a force on the current we define as a MAGNETIC FIELD.

As you can see there is an analogy between electric and magnetic fields

There are no magnetic charges and so we can not use them to study the magnetic field.

That is why magnetic field is studied with a small loop or frame of wire with a current.

That loop must be hanged by the elastic wires. When the current is flowing though the

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loop the magnetic forces create a torque. The greater the torque the more the angle of

rotation. By measuring the magnitude of that angle we can estimate the strength of the

magnetic field.

The same loop can be

used to discover the direction

of the field. As we studied

before in mechanics, a torque

is the product of the applied

force and lever arm. On a

diagram (a) the lever arm is

nonzero, so the torque ≠ 0 and

the loop rotates. If the

connected twisted wires are non-elastic (the loop rotates freely), then loop will rotate

until its lever arm and consequently the torque will become zero (b). Scientists agreed to

define the direction of magnetic field as the direction of the perpendicular to the surface

of such stopped loop. That loop should be initially suspended on non-elastic twisted

wires.

Because every loop has two perpendiculars the corkscrew rule had been established.

If the corkscrew’s handle rotates as current directed than the direction of its

motion will coincide with the direction of the magnetic field.

Another way to discover the direction of the field is to use a compass or a magnetic

arrow. If we place it into the field, it starts to rotate and finally comes to rest. Then the

direction from the S pole toward the N pole is the direction of the magnetic field.

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If we place different loops with different currents at the same point of the

magnetic field and measure the maximum torque applied on them, we will observe that

the ratio of the torque over the current vs. area product is equal for each loop. That means

that that ratio depends only on the field properties and it can be used as the characteristic

of the magnetic field. (Analogous to the electric field intensity). That ratio is defined as

magnetic induction.

Magnetic induction is the ratio of the maximum torque exerted on a loop with

current at a given point of the magnetic field over the product of current times

loop’s area.

Where B is the magnetic induction, - torque, exerted on a loop

I - the current, and A is the area of the cross-section of the loop.

The unit of the magnetic field induction is Tecla (T) - it is the induction of the magnetic

field, where a torque of 1N • m is exerted on a loop with the current of 1 A and with the

cross-section of 1 m2. Another unit that is widely used in measuring the magnetic

induction is Gauss. Gauss was the unit of magnetic induction in CGS system units, a

system based on centimeter, gram and second. That system was in use before scientists

decided to use SI system.

1 Gauss = 10-4

Tecla. In problems gauss should be changed to Tecla.

MAGNETIC FIELD LINES

These lines do not exist in nature. Physicists to visualize magnetic fields invented them.

By the definition the magnetic field line is the imaginary line, the tangent of which at

any point is always directed as the induction of a magnetic field at the same point.

We can observe those lines with iron filings. Several examples of magnetic fields, you

can see below.

The field of the bar magnet. The field of a straight current

B max

IA

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2 scientist Biot and Savart had discovered the formula for the magnetic induction for the

field of a straight current. They derived that the induction of magnetic field created by the

infinite straight current at a distance R from the wire can be computed by formula.

where µ is called the magnetic permeability. That coefficient describes the magnetic

properties of the matter. Its analogy in electric field is dielectric constant . For vacuum or air µ = 1.

µo-- is the permeability constant. It equals 1.26x10-6

H/m.

The direction of the magnetic field created by the straight current can be discovered by

using two similar rules.

Method 1: The corkscrew rule:

If the corkscrew moves the same

direction as the current, than the

direction of the handles rotation

coincides with the direction of the

magnetic field

If you want to know the direction of

the corkscrew motion you can use a

rule Rightly Tightly, Lefty

Loosely. (This rule was suggested

to me by one of the students)

Method 2: The Right hand rule. If

the thumb of your right hand points

the direction of current in a straight

wire, than your bent fingers point

the direction of the magnetic field lines.

The field of the coil of wire (solenoid) with a current.

It is interesting to observe

that the shape of the

magnetic fields for a bar

magnet and a coil of wire

are the same. We will use

that idea later. The

induction of the of the magnetic field (magnetic field strength) inside the solenoid can be

expressed as

where n is the number of turns per unit of length.

B o

I

2R

B onI

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Forces on a current in a magnetic field (Ampere’s law)

If we place a conductor with a current into magnetic field a force would be exerted on the

current. Ampere discovered that the force is equal to

F=IBL sin

where I- is the current, B is the magnetic induction, L-is the wire’s length and is an

angle between the directions of the magnetic induction and the current.

The direction of the force can be determined by a left-hand rule.

Point the fingers in the direction of the current (conventional current); position your

palm in such a way that the magnetic lines are directed into it. Then the unbended

thumb would give you the direction of the force acting on the wire.

Example:

In several textbooks you can discover a different right hand rule.

You have to orient your right hand so that the outstretched fingers point in the

direction of the conventional current. If you bend your four fingers they should be

directed as a magnetic field. Then the unbended thumb would give you the direction

of the force acting on the wire.

Which rule you prefer it is your option.

The phenomenon of the existence of the magnetic forces is widely used in various

electrical devices, for example in galvanometers and electric motors.

Example: Let us place a rectangular loop in a magnetic field created by two

permanent magnets. If we apply the left-hand rule we discover that the downward force is

exerted on wire CD and upward force is exerted on wire AB. As for the wires BC and

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DA - there is no force exerted, because current in those wires is parallel to the direction

of the magnetic field. ( = 0, sin =0 and thus the force = 0).

INTERACTION OF THE PARALLEL CURRENTS

Let assume that we have two long parallel wires with

currents of I1 and I2 respectively located at a distance of

R and we want to estimate the force between them.

We should assume that the second wire is located in a

magnetic field created by the first one.

According to the Biot- Savart law the field created by the current 1 at a distance R is

According to the Amphere’s formula the force exerted on the current in a magnetic field

is

F=I2BL

If we substitute B into F, the formula for the force between two currents would be

Motion of the Single Charged Particle in Magnetic Field.

The force exerted on a current could be represented as the sum of the forces exerted on

all charged particles, which created those currents. Then the force exerted on a single

particle has to be N times less then the force exerted on the current, where N is the

number of those particles inside the wire.

As we know the current can be expressed as

I =qvAn,

where q- is the charge of a single particle, v is the average particle’s velocity,

A- the wire’s cross-section area, n is the particle’s density, i.e. the number of the particles

per unit volume.

If we substitute that expression into the expression for force and also assume that

= 90° (the particle is moving perpendicular to the magnetic field), we have

It is easy to understand, that AL =V, where V- is the volume of the wire, and that the

nV=N. That is why we can eliminate nAL from the top and N from the bottom. Finally we

have

B o

I1

2R

F BILsin

N

F BnAvL

N

F o

I1I2

2RL

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F= Bvq

This is the expression for the force exerted by the magnetic field on a single charged

particle that is moving inside that field.

That force is often called the Lorentz force

As for the force exerted on a current the direction of the force exerted on a single charged

particle can be found by the left-hand rule. The only difference is that instead of directing

fingers along the current, we have to direct them along the direction of motion for

positively charged particles and against the direction of the velocity for the negatively

charged particles.

The unbended thumb is always perpendicular to the fingers. That means that the direction

of the force always perpendicular to the velocity. The force that is perpendicular to the

velocity creates the centripetal acceleration. So a particle will have a circular trajectory.

We can easily find the radius and the period of rotation.

According to the second Newton’s law

F=ma

F is the force exerted on a particle F=Bvq, a is the centripetal acceleration = V2/R, where

R is the radius of the rotation. So, we have

The period can be computed as 2πR/v - the circumference over the velocity. We have

As we can see the period does not depend on the velocity. That property is used in

accelerators of the elementary particles.

Electromagnetic Induction

Let us run several experiments.

1. A coil with wire is connected

to a Galvanometer.

If the permanent magnet is at rest,

then there is no current in a coil. If

the magnet is in motion, then the

current flows. Its magnitude

depends on the magnet velocity.

Its direction depends on the

direction of the magnets motion

and polarity.

Bvq mv2

R or R =

mv

qB

T 2R

v

2m v

qB

v

2m

qB

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2. A coil with a current is moving relatively to another coil, connected with an

amperemeter.

3.

Analyzing all data we come to a conclusion, that when a permanent magnetic

field is penetrating through the closed circuit, there is no current in the circuit. If the flux

is variable, then the current appears.

The process of the current generation by a variable magnetic field is defined as the

ELECTROMAGNETIC INDUCTION.

MAGNETIC FLUX

As we can observe the effect of electromagnetic induction takes place when the number

of the magnetic field lines through the closed circuit varies. To describe that number

physicists use a new quantity - magnetic flux.

Let us assume that the loop with the area of A is located in a magnetic field with

induction of B and that the angle between the direction of the field and the normal to the

loop is .

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Magnetic flux is

defined as the product

of the magnetic field

induction times the

area of the loop times

cosine of an angle

between the direction

of the field and the loop’s

normal.

= B A cos

The unit of the magnetic

flux is Weber = T*m2.

Magnetic flux through the

loop is at maximum when

the normal is parallel to

the filed (pic. 1). Magnetic

flux through the loop is 0

if the loop’s normal is

perpendicular to the direction of the magnetic field. (Pic.2)

FARADAY'S LAW

Electromagnetic induction can be described with the electromotive force created inside

the closed circuit in the time of the effect. As we mentioned before, the magnitude of the

current is proportional to the rate at what the magnetic field does change.

That rate can be described with a quantity

Where - is the magnetic flux, which is equal to BA.

Let us run dimensional analyses of that quantity.

That quantity is measured in Volts and that is why it can be used for the EMF

measurements. Thus, we have

Sign minus is introduced because of the Len’z law that we will discuss below.

t

t

BA

t

Tecla m2

s

Newton

A m m

2

s

Newton m

A s

Joule

Coulomb Volt

= -

t

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LENZ’S LAW

The goal of the Lenz’s research was to investigate the direction of EMF and thus the

direction of the induced current.

To obtain that goal let us run an imaginary experiment.

We move a bar magnet toward the coil.

The magnetic flux through the coil increases. That variable flux created the EMF

and the induced current starts to flow through the coil.

That current creates its own magnetic field. We do know that the magnetic field of the

wire coil with the current has the same shape as the field of the bar magnet. That means,

that on the edge B of the coil we have one pole and on the edge C another one.

Let us initially assume that S pole is located at the edge B of the wire coil. That means

that if we initially push the magnet somewhere far away from the coil, it will infinitely

continue to move, because of the attraction between the unlike poles and that the current

will be created without any energy input. That is impossible, because it contradicts to the

energy conservation law. So, we have to have pole N on the edge B.

Than, if you move the magnet toward the coil, you have to do some work to overcome

the repulsion between the like poles. That work is transforms into the current’s energy.

If we draw the magnetic field lines for the magnetic field created by the induced current,

we can observe that that field opposes the increase of the flux through the coil.

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If we move the magnet out of the coil, then the flux through the wire coil decreases. We

can repeat our reasoning and prove that the pole S has to appear on the edge B.

That time the magnetic field created by the induced current opposes the decrease of the

flux through the coil.

In general Lenz’s law states: The current is induced in a direction such that the

magnetic field produced by the current oppose any change in flux that induced the

current.

We can observe the effects of the Lenz’s law with the real experiment.

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If we move the magnet toward the aluminum ring, it moves away. If we move the magnet

out of the ring, it follows the magnet.

Explanation: When the magnet moves toward the ring the flux through the ring increases

and the induced current create a magnetic field that tries to prevent that increase. So it

moves the ring out of the magnet.

When the magnet moves away from the ring the flux through the ring decreases and the

induced current create a magnetic field that attempts to prevent that decrease. Thus, the

ring follows the magnet.

EMF in a straight wire

Let us assume that a straight wire with a length L is moving in a magnetic field with the

velocity V. We also assume, that the direction of the magnetic field is normal and out of

the sheet of paper.

Magnetic field does exert a force on each electron

in a moving wire. According to the left-hand rule

that force is directed toward point B. so all

electrons would move toward the edge A and it

becomes negative. At the same time the lack of

electrons on edge B will create a positive charge

there.

Let us assume that at time ∆t, the wire will travel

distance d=v∆t. It will cross all magnetic lines in a

rectangle ABA*B*. So the magnetic flux change is

and the EMF is equal

Self-Inductance

Let us run an experiment. We design a circuit and turn the power on.

BA BLd BLvt

=

t

BLvt

t BLv

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We do observe that bulb 2 starts to glow immediately and bulb 1 experiences a certain

delay. The explanation is simple. When we turn the current on, it starts to rise and it

creates the rising magnetic flux through the coil. That variable flux will create an induced

current. The induced current according to the Lenz’s law resists to that rise and so we

observe the delay in lighting of the bulb 1.

We can observe the similar effect when we turn the current off. The schematic of

the experiment is in the diagram below.

When we turn the current off, decreasing current creates decreasing magnetic flux. That

flux pierces the coil and creates the induced current. That current according to the Lenz’s

law resists the initial decrease of the main current. So, it has the same direction in the

coil, but opposite direction through the ammeter. That effect we can observe.

The phenomenon, when the current itself creates the EMF that resists to the current

change is called self-inductance.

There is no difference in principal between the effects of electromagnetic induction and

self-inductance. It is more question of terminology. In the effect of electromagnetic

induction, the EMF is created as the result of the change of the external magnetic field.

In the effect of self-induction the EMF is created by the change of the internal magnetic

field, i.e. created by its own current. During the effect of self-induction the magnetic flux

through the conductor is proportional to the current through it.

= L I

L is the coefficient that depends on the geometry of the conductor (for example in the

case of the coil it depends on the number of turns and on the cross-section area) and on

the surrounding medium. That coefficient is called an inductance.

the unit of inductance is Henry (H)

Henry = Weber/Amp

As we know, according to Faraday’s law, the EMF = ∆ / ∆t. If instead of we

substitute LI, we obtain the final formula for the EMF of the self-inductance

self = -

t L

I

t

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ENERGY OF A MAGNETIC FIELD

Let us compare two effects. Inertia in mechanics and self-induction in magnetism.

Inertia describes the body's unwillingness to change its velocity. Self-induction describes

the conductor's unwillingness to change current through itself.

That led us to conclusion that those effects are analogous.

That means that velocity (v) is analogous to the current (I), and mass (m) is analogous

to the inductance (L).

As we know in mechanics the body's energy can be expressed as mV2/2.

In analogy the energy of the magnetic field, created by current can be expressed as

LI2/2.

To prove that our formula is correct, we can complete dimensional analyses

Home assignment:

Cutnell: Chapter 21 Conceptual questions: page 659… #1,3,4,5, 6,8, 9, 14, 16, 19

Problems; Page 661 #3, 7, 8, 9, 15, 19, 21, 23,24, 29,31,33, 34,42, 43, 46

53, 54,56

Cutnell: Chapter 22 Conceptual questions: page 699… # 4, 6,7, 10,12

Problems; Page 700 #3, 5,7,15,16, 24, 25,26,32, 33,46,

E LI2

2

Henry Amp2

Weber Amp2

Amp Tecla m

2Amp

Newton m

Amp m2 m

2Amp Joule

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