Section III Em i

8/7/2019 Section III Em i

1/19

Lecture Notes Magnetism to Amperes Law

By. S.M Alay-e-Abbas

1

Magnetic Field

It turns out the hardest thing to understand about magnetism is a simple magnet.

We will start by studying the force on a current caused by a magnet field. We'll

wait until next sections to figure out where the magnetic field comes from.

The Force Between Conductors

There are similarities between the electric force and the force between wires.

Could the force between wires just be the electric force? It is not because:

1) Like currents attract and opposite repel, exactly the reverse of the electric force.

2) The current carrying wires are electrically neutral. They exert no force on a

single charge.

It would seem that we must treat the force between current carrying wires as a new

force called the "magnetic force." It must be noted that, in fact, this force is

electrical in nature and Einstein's Theory of Relativity explains the connection

between electricity and magnetism.

We need to establish the force law (analogous to Coulomb's Rule or Newton's Law

of Universal Gravitation) for this "new" force of magnetism.

Newtons law requires that F12 = F21. You must guess that


2/19



2

and

and

and

Combining these facts we get

Where the constant of proportionality is

The definition of Magnetic Field

Recall the way we defined electric field. In that case instead of thinking q1 exerting

a force on q2 we thought of q1 creating a field and q2 feeling the force due to being

in this field.

This Implied

where F21 = q2E


3/19



3

We can do the same thing with magnetic field, defining it with the Alphabet B.

Where F21 = I2lB1

Where B1 = (uo/2) (I1/r)

To incorporate the vector nature of forces we need to pick a direction for the

magnetic field. Since F is up and l is in the horizontal plane along the wire, it is

most convenient to choose B into the paper. Now we can define the magnetic field

vector as,

Note that

It is convenient to define unit of B as

Another unit is

Currents are the source of the magnetic field. We will discuss these facts in detail

in the next topics. These sections will focus on the effect of an applied magnetic

field.


4/19



4

The magnetic Force on a Moving Charge

The force on a current carrying wire is

The current is composed of individual charges. We want to know the magneticforce on a single charge. Recall the definition of current density,

Using the expression for drift velocity

Inserting the free electron density

The force on one charge will be


5/19



5


6/19



6

Current Loops in Constant Magnetic Field

A uniform magnetic field points along the z-direction. An arbitrarily shaped

current loop is placed in the field.

Let's find the net force on the loop. The force, dF, on a small segment of the loop,

dl, is given by the definition of magnetic field

The total force on the loop is

An arbitrary length is given by

And the magnetic field

The cross product is

The total force on the loop is

The above relation is true for any shape of current loop in electric field.

But what is the torque on the loop. In books you may find only square loops but we

are considering a circular loop as shown below.


7/19



7

The loop will be in the x-y plane and B is in the x-z plane and angle from the z-

axis. This geometry means no loss of generality.

The definition of torque is

So the torque element about the center of the center of the loop on a small

segment caused by will be

Figure below shows the relations of different variable involved in this study

The vectors that we need are

and

Now


8/19



8

and

This gives

If we make the following definition

Than the torque is

This equation is similar to the result for the torque on an electric dipole. By using

this analogy we can write the potential energy of a magnetic dipole in a magnetic

field as,


9/19



9

Magnetic Devices

Velocity Selector

Particles of mass m and charge q move at a speed u into a region with a vertical E-field and a horizontal B-field as shown. It turns out that there will be only one

velocity that will allow the particles to be undeflected by the fields. It can be found

by using the Second Law,

Apply the definition of E-field and the magnetic force on a moving charge,

For the undeflected particles,


10/19



10

Mass Spectrometer

Galvanometer

Hall Probe


11/19



11

the right

Cyclotron


12/19



12

Stereo Speaker


13/19



13

Magnetic Field Due to Currents

Now that we know the effects of magnetic fields on currents we can discuss the

causes of magnetic fields. If currents feel the magnetic force, Newton's Third Law

requires they must exert it as well. The purpose of these sections is to learn to

calculate magnetic fields caused by currents. The Biot-Savart Rule is a recipe that

will always work. Amperes Law, while more fundamental, can be used to find

fields only in cases where the current distribution is highly symmetric.

The Bio-Savart Law

The electric force on a charge q1 is found by calculating the field due to all the

other charges. We start by finding the field due to the point charge dq2 i.e.

Next the force on q1 due to the charge dq2 is found by the expression

Then we integrate the contribution of forces due to all the other charges by

considering the charge distribution.

The same procedure will be followed to find the force on a current I1 due to

distribution of current I2. The magnetic force on the current I1in the magnetic field

dB2 caused by the current element I2 is


14/19



14

Comparing equivalent things in electric and magnetic force equations we get.

Using an analogy to find the field due to electric charge element dq is

This is called Biot-Savart Law.

Where the magnetic constant is

It is convenient to know that the magnetic field due to long straight wire is


15/19



15


16/19



16

Amperes Law

Amperes Law states that the sum of the magnetic field along any closed path is

proportional to the current that passes through. Mathematically.

In order to understand the idea consider the circular path around a current carrying

wire that passes through the center of the path and is perpendicular to the plane of

the path

We know that the magnetic field due to this wire is

and is parallel to

So we have

Which is consistent with Amperes Law.

Like Gausses Law Amperes Law is also used to find Magnetic field due to

sufficiently symmetric charge distributions.

Integral and Differential Forms

According to Amperes law


17/19



17

==

=

=

=

s

o

c

o

s

n

c

o

dAjIdsB

Hence

dAjI

Since

iI

where

IdsB

..

.

.

The last relation is called integral form of Amperes law.

Now integral for of Amperes law is

= so

cdAjdsB ..

using Stokes Theorem we convert the line integral on the left hand side into

surface normal integral as

jcurlB

Hence

dA

ce

jcurlB

gives

dAjcurlB

dAjdAcurlBdsB

o

o

s

o

sc s

o

=

=

=

==

0

sin

0

0].[

...

Which is the differential form of Amperes law it is also one of the basic

Maxwells equation of electromagnetism.


18/19



18


19/19



19

Date post:	08-Apr-2018
Category:	Documents
Upload:	asif-rasheed
View:	222 times
Download:	0 times

Section III Em i

Documents