Gradient of Scalar Gradient of Scalar FieldField

In Cartesian co-ordinates:In Cartesian co-ordinates:

^ ^ ^

x y zx y z

ff ff

¶ ¶ ¶Ñ = + +

¶ ¶ ¶

Gradient of Scalar FieldGradient of Scalar Field

Gradient of Vector FieldGradient of Vector Field

Divergence of a vector field A is defined as the net outward flux of A per unit volume as the volume about the pointtends to zero, i.

0

.

. lim s

AdS

An n®

Ñ =D

òV

Ñ

Gradient of Vector FieldGradient of Vector Field

The flux of a vector field (the integral term The flux of a vector field (the integral term inin

this equation) is analogous to the totalthis equation) is analogous to the totalflow of an incompressible fluid through aflow of an incompressible fluid through asurface (surface (S S in the equation). For a closedin the equation). For a closedsurface, the total flow through the surface surface, the total flow through the surface

isisOften zero - if Often zero - if AA represents water flow, for represents water flow, forexample, then the total amount of waterexample, then the total amount of waterflowing intoflowing into a surface is always matched a surface is always matchedby the by the amount coming outamount coming out..

Problem of Gradients:Problem of Gradients:Equipotential lines change by Equipotential lines change by 2 v m2 v m-1-1 in the in the

xx direction and direction and 1v m1v m-1-1 in the in the yy direction. direction. The potential does not change in The potential does not change in z z direction. Find the electric field direction. Find the electric field EE

^ ^

( )v v

E v x yx y

¶ ¶=- Ñ =- +

¶ ¶

^ ^0 12 1 2.24 26.6x y vm-+ = Ð

2 2 1 012 1 2.24 ; tan ( ) 26.6

2q -+ = = =

Electric FiledElectric Filed

31 22 2 2

0 1 2 3

1( )

4

QQ QE

r r rpe= + +

3

210

1

4N

N N

QE

rpe =

= å

Superposition of Electric Field

http://physics-help.info/physicsguide/electricity/electric_field.shtml

Electric FieldElectric FieldThe Superposition PrincipleThe Superposition Principle for charge for charge

continuously distributed over an object continuously distributed over an object line charge

Surface

Volume

where:   is radius position of point where the electric field is defined with respect to small volume dV with volume charge density V is volume of the charged object

rr

Electric FieldElectric FieldElectric field produced by uniformly charged ring at its axis

with magnitude

where:

is radius position of point P where the electric field is defined with respect to center of the ring R is radius of ringQ is total charge of the ring

rr

Electric FieldElectric Field

Volume with Volume with SourceSource

Source:-Divergence is

Non-Zero at the lit point.

Volume with Volume with SinkSink

To imagine what a To imagine what a sinksink would lookwould look

Like; Imagine video-taping theLike; Imagine video-taping the

sparkler and replaying it backwards.sparkler and replaying it backwards.

In that case the sparks would flowIn that case the sparks would flow

back into the lit point – this wouldback into the lit point – this would

then appear to be a then appear to be a sinksink..

DivergenceDivergenceThe net outward flow The net outward flow

per unit volume isper unit volume is

therefore a measure of therefore a measure of the the

strength of the strength of the enclosed enclosed

source (source (note this is a note this is a

scalar quantityscalar quantity). In ). In

Cartesian co-ordinates Cartesian co-ordinates

the expression for div A the expression for div A

is:is:

. yx zAA A

Ax y z

¶¶ ¶Ñ = + +

¶ ¶ ¶

Divergence of Magnetic Divergence of Magnetic FiledFiled

• B = 0

The interpretation of this zero divergence

relation is that no sources or sinks

of magnetic charge exist.

Divergence of Electric Divergence of Electric FiledFiled

Electric FieldElectric Field

• D = v

The electric flux density D, it can be shown that

Where D is Flux Density , v is the free electric charge density

Divergence & Curl of Divergence & Curl of FieldsFields

visualize the fields as resulting from forces generated by whirlpool-like vortices in the ether.

Maxwell was therefore interested in the rotation (or circulation, or “vorticity”)of the fields which today we represent in terms of the curl of the field vectors.

Curl of FieldsCurl of Fields

Curl Curl E = ∂ B/ ∂tE = ∂ B/ ∂t

which describes the generation of a “vortex” in the electric field

resulting from a time varyingmagnetic field.

Curl of FieldsCurl of Fields

Curl B = Curl B = μμE + E + μμ ∂ E

∂t

µóE term describes the generation of avortex in the magnetic field resulting from the passage of an electric current through the medium.

The second term represents a further contribution to the vortex resulting from a time-varying electric field.

Faraday CageFaraday Cage

A human-size Faraday cage which allows someone to stand inside while the discharge from a Tesla coil is directed towards it. The person inside is asked to hold a fluorescent tube, which does not light.

While similar tubes balanced against the sides of the cage do light when the discharge is enabled.

Faraday DiscFaraday Disc

First Dynamo 1831First Dynamo 1831

First dynamo ever invented, described by Faraday in Experimental Researches in 1831. The device consists of a copper disc that can be rotated in the narrow gap between the poles of a magnet. Contacts near the centre and edge of the disc (made more conducting by the application of mercury) develop a voltage between them

Faraday DiscFaraday DiscGeneratorGenerator

BMagnetic filed varies with time and is given by

B = B0 sint, where B0 is maximum Flux density

Example 1 Faraday LawExample 1 Faraday Law• A time-varying magnetic filed is given by:A time-varying magnetic filed is given by:

0 cos yB B t aw=Where B0 is constant . It is desired to find the induced emf around the rectangular loop C in the xz- plane bounded by the lines x=0, x = z , z=0, and z = b. as shown in figure:

x

z

b

a

OChoosing dS = dx.dzay according to the right hand rule the Total flux enclosed by the loop is therefore:

00 0

0 00 0

. cos .

cos cos

b a

y ys

b a

B dS B ta dxdza

B t dxdz abB t

f w

w w

= =

= =

ò ò ò

ò ò

Example 1 …..Example 1 …..

00 0

0 00 0

. cos .

cos cos

b a

y ys

b a

B dS B ta dxdza

B t dxdz abB t

f w

w w

= =

= =

ò ò ò

ò ò

abB0

abB0

emf

[ ]0 0

. .

cos sin

sc

dE dl B dS

dt

dabB t abB t

dtw w w

=-

=- =

ò òÑ

Example 2 Faraday LawExample 2 Faraday Law

dS

y

x

A rectangular loop of a wire with three sides fixed and the forth side movable is situated in a plane perpendicular to a uniform magnetic field B=B0az as shown in the figure. Moving side is a conducting bar moving at the velocity of v0 in y- direction. It is desired to find emf. generated around the loop C.

The position of the bar at any time t is y0+v0t and considering dS=dx.dy az

v0ay

0 0

0

0

0 0

0 0 0

. .

( )

z z

s s

y v tl

B ds B a dxdya

B dxdy

B l y v t

+

=

=

= +

ò ò

ò ò

C

Example 2 …..Example 2 …..

dS

y

x v0ay

0 0 0

0 0

. .

[ ( )]

s s

dE dl B dS

dt

dB l y v t

dtB lv

=-

=- +

=-

ò òÑ

Note that once the bar is moving to the right, the induced e.m.f. is negative and produces a current in the sense opposite to that of C. This polarity of the current is such that it gives rise to a magnetic field directed out of the paper.

C