emt formula sheet bsc level

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

Website: www.physicsbyfiziks.com Email: [email protected] 1

Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16

ELECROMAGNETIC THEORY FORMULA SHEET

CONTENTS

PART-A (CORE)

1. Electrostatics…………………………………………………………………..... (5-17)

1.1 Coulomb’s Law and Superposition Principle…….................................................. (5)

1.1.1 Electric field

1.2 Gauss’s law ……..................................................................................................... (6)

1.2.1 Field lines and Electric flux

1.2.2 Applications

1.3 Electric Potential……............................................................................................. (7)

1.3.1 Curl of Electric field

1.3.2 Potential of localized charges

1.4 Laplace’s and Poisson equations…….................................................................... (8)

1.5 Electrostatic boundary condition…….................................................................... (8)

1.6 Work and Energy in electrostatics…….................................................................. (9)

1.6.1 The energy of point charge distribution

1.6.2 Energy of continuous charge distribution

1.7 Basic properties Conductors ……........................................................................ (10)

1.8 Multiple Expansions……..................................................................................... (11)

1.8.1 The Electric Potential and Field of a Dipole

1.8.2 Approximate potential at large distances

1.9 Polarization……................................................................................................. (13)

1.9.1 The Field of a polarized Object (Bound Charges)

1.10 The Electric Displacement............................................................................. (13-14)

1.10.1 Gauss Law in the Presence of Dielectrics

1.10.2 Linear Dielectrics (Susceptibility, Permittivity, Dielectric Constant)

1.10.3 Boundary Condition

1.10.4 Energy in dielectric system





1.12 Image problems.................................................................................... (15-17)

1.12.1 The Classic Image problem

1.12.2 Induced Surface Charge

1.12.3 Force and Energy

1.12.4 Other Image Problem

2. Magnetostatic.......................................................................................... (18-24)

2.1 Magnetic force on current element............................................................. (18)

2.1.1 Current in a wire

2.1.2 Surface current density

2.1.3 Volume current density

2.2 Continuity equation................................................................................... (19)

2.3 Biot-Savart law.......................................................................................... (19)

2.3.1 Magnetic field due to wire

2.3.2 Magnetic field due to Solenoid and Toroid

2.4 Ampere's Law........................................................................................... (20)

2.5 Magnetic Vector Potential........................................................................ (20)

2.6 Magnetostatic boundary condition........................................................... (21)

2.7 Multiple Expansion of Vector Potential................................................... (22)

2.8 Magnetisation........................................................................................... (23)

2.8.1 The Field of a magnetized Object (Bound Currents)

2.9 The Auxiliary field H.......................................................................... (23-24)

2.9.1 Ampere’s Law in in presence of magnetic materials

2.9.2 Magnetic Susceptibility and Permeability

2.9.3 Boundary Condition

3. Dynamics of charged particles in static and uniform electromagnetic fields

3.1 Charged particle in static electric field………………………..……. (25-26)

3.1.1 Charged particle enters in the direction of field (Linear motion)

3.1.2 Charged particle enters in the direction perpendicular to field

(Parabolic motion)





3.2 Charged particle in static magnetic field.......................................................... (27)

3.2.1 Charged particle enters in the direction perpendicular to field

(Circular motion)

3.2.2 Charged particle enters in the direction making an angle with the field

(Helical motion)

3.3 Charged particle in uniform electric and magnetic field (Cycloid motion).... (28)

4. Electromagnetic induction…………………………………………………... (29)

4.1 Faraday’s Law

4.1.1 Lenz’s Law

4.1.2 Inductance

4.1.3 Energy stored in the field

5. Maxwell's equations…………………………………………………..…... (30)

5.1 Maxwell’s equation in free space

5.1.1 Electrodynamics before Maxwell’s

5.1.2 How Maxwell fixed Ampere’s Law

5.1.3 Paradox of charging capacitor

5.1.4 Maxwell’s equation in free space

5.2 Maxwell’s equation in linear isotropic media …………………………... (30)

5.3 Boundary conditions on the fields at interfaces ……………………….... (31)

6. Electromagnetic waves in free space…………………………….….... (31-34)

6.1 Poynting Theorem…………………………………………………..….... (31)

6.2 Waves in one dimension (Sinusoidal waves) ……………….................... (32)

6.2.1 The wave equation

6.2.2 Terminology

6.2.3 Complex notation

6.2.4 Polarization





6.3 Electromagnetic waves in vacuum……………….................................... (35)

6.3.1 The wave equation for E and B

6.3.2 Monochromatic plane waves

6.3.3 Energy and momentum in Electromagnetic wave

6.4 Electromagnetic waves in matter………………..................................... (38)

6.5 Electromagnetic waves in conductors……………….............................. (39)

7. Applications of Electromagnetic waves………................................ (41-43)

7.1 Reflection and Refraction………………................................................ (41)

7.1.1 Normal incidence

7.1.2 Oblique incidence

7.1.3 Fresnel’s relation (Parallel and Perpendicular Polarization)

8. Potential and Field formulation for Time Varying Fields……........ (44-45)

8.1 Scalar and vector potentials..................................................................... (44)

8.2 Gauge transformation.............................................................................. (44)

8.3 Coulomb and Lorentz gauge.................................................................... (45)





1. Electrostatics

The electric field at any point due to stationary source charges is called as electrostatic

field.

1.1 Coulomb’s Law and Superposition Principle

The electric force on a test charge Q due to a single point charge q, which is at rest and a

distance R apart is given by Coulomb’s law

20

14

QqFR R

.

The constant 0 is called the permittivity of free space.

In mks units, 2

120 28.85 10

.C

N m

1.1.1 Electric field

If we have many point charges 1 2, ,......q q at distances 1 2, , ......R R from test charge Q,

then according to the principle of superposition the total force on Q is

1 21 2 2 2

0 1 2

1............ ........1 24q Q q QF F FR RR R

,

F QE

where 210

14

ni

ii i

qE P RR

E

is called the electric field of the source charges. Physically E P

is the force per unit

charge that would be exerted on a test charge placed at P.

If charge is distributed continuously over some region, then

20

1 1 ˆ4 line

E r RdqR

.

The electric field of a line charge is dq dl

20

1 ( ) ˆ4 line

rE r RdlR

where is charge per unit length.

Q

q

R





For surface charge dq da

20

1 ( ) ˆ4 surface

rE r RdaR

where is charge per unit area.

For a volume charge dq d

20

1 ( ) ˆ4 volume

rE r RdR

where is charge per unit volume.

1.2 Gauss’s law

1.2.1 Field lines and Electric flux

Consider that a point charge q is situated at the origin: 20

14

qEr r

This field is represented by the field line as shown in

figure below. The magnitude of the field is indicated by

the density of the field lines: it's strong near the center

where the field line are close together, and weak farther

out, where they are relatively far apart.

The field strength (E) is proportional to the number of

field lines per unit area (area perpendicular to the lines).

The flux of E through a surface S, .ES

E da

is a measure of the “number of field

lines” passing through S.

A charge outside the surface would contribute nothing to the total flux, since its field

lines go in one side and out the other. It follows, then, that for any closed surface,

0

1. encE d a Q

where encQ is the total charge enclosed within the surface. This is Gauss’s law in integral

form.

Gauss’s law in differential form: 0

1.E

E





1.2.2 Applications of Gauss’s Law

Gauss's law is always true, but it is not always useful. Gauss's law is useful for only three

kinds of symmetry:

1. Spherical symmetry. Make your Gaussian surface a concentric sphere.

2. Cylindrical symmetry. Make your Gaussian surface a coaxial cylinder.

3. Plane symmetry. Make your Gaussian surface a “pillbox,” which extends equally

above and below the surface.

1.3 Electric Potential

1.3.1 Curl of Electric field

Then integral around a closed path is zero i.e. . 0E d l

This line integral is independent of path. It depends on two end points.

Applying stokes theorem, we get 0E

. The electric field is not just any vector but

only those vector whose curl is zero.

So, we can define a function .P

V r E d l

where is some standard reference point. V then depends only on the point r. It is called

the electric potential.

Evidently, the potential difference between two points a and b is

.b

a

V b V a E d l

E V

Potential obeys the superposition principle.

Gaussian surface

Gaussian pillbox

Rr

surfaceGaussian





1.3.2 Potential of localized charges

Potential of a point charge q is0

14

qVR

where R is the distance from the charge.

The potential of a collection of point charge is10

14

ni

i i

qVR

.

For continuous volume charge distribution0

1 ( )( )4

rV r dR

The potential of line and surface charges are 0

1 ( )( )4

rV r dlR

and0

1 ( )( )4

rV r daR

.

1.4 Laplace’s and Poisson equations

Since E V

and 0

.E

2

0

V

.

This is known as Poisson's equation.

In regions where there is no charge, so that 0 , Poisson's equation reduces to Laplace's

equation, 2 0V .

1.5 Electrostatic boundary condition

The boundary between two medium is a thin sheet of surface charge .

0

above below

and || ||above belowE E

0

ˆabove belowE E n

where n̂ is unit vector perpendicular to the surface, pointing upward.

above belowV V 0

above belowV Vn n

where nVnV ˆ denotes the normal derivative of V (that is the rate of change in the

direction perpendicular to the surface.)





1.6 Work and Energy in electrostatics

The work done in moving a test charge Q in an external field E , from point a to b is

b b

a a

W F dl Q E dl Q V b V a

If a and b r

W Q V r V QV r since 0V

In this sense potential is potential energy (the work it takes to create the system) per unit

charge (just as the field is the force per unit charge).

1.6.1 The energy of point charge distribution

When the first charge 1q is placed, no work has been

done. When 2q is placed work done 2 2 1W q V

where 1V is the potential due to 1q so,

12 2

0 12

14

qW qR

.

Similarly when third charge 3q is placed

1 23 3

0 13 23

14

q qW qR R

.

The work necessary to assemble the first three charges is 1 3 2 31 2

0 12 13 23

14

q q q qq qWR R R

.

In general, 1 1 1 1 10 0

1 1 14 8 2

n n n n ni j i j

i ii j i j iij ij

j i j i

q q q qW qV r

R R

,

where V(ri) is the potential at point ri (the position of qi) due to all other charges.

1q

2q iq

Q

a

b

1q1r

2r 2q

3q

3r

13R

12R

23R





1.6.2 Energy of continuous charge distribution

For a volume charge density 12

W Vd and 20 2 all space

W E d

(i) Energy of a uniformly charged spherical shell of total charge q and radius R is

2

0

qW8 R

.

(ii) Energy stored in a uniformly charged solid sphere of radius R and charge q is

2

0

1 3qW4 5R

.

1.7 Basic properties Conductors

1. 0E

inside a conductor.

2. 0 inside a conductor.

3. Any net charge resides on the surface.

4. A conductor is an equipotential.

5. E is perpendicular to the surface, just outside a conductor.

Because the field inside a conductor is zero, boundary condition

0

ˆabove belowE E n

requires that the field immediately outside is0

ˆE n

.

In terms of potential equation 0

above belowV Vn n

yields 0

Vn

These equations enable us to calculate the surface charge on a conductor, if we can

determine E

orV .

Force per unit area on the conductor is 2

0

1 ˆ2

f n

.

This amounts to an outwards electrostatic pressure on the surface, tending to draw the

conductor into the field, regardless the sign of . Expressing the pressure in terms of the

field just outside the surface, 20

2P E .





1.8 Multiple Expansions

1.8.1 The Electric Potential and Field of a Dipole

If we choose coordinates so that p

(dipole moment) lies at the

origin and points in the z direction, then potential at ( , )r is:

2 2

.̂ cos,4 4dip

o o

r p pV rr r

.

30

1 ˆ ˆ, 34

dipE r p r r pr

Note:

(a) When a dipole is placed in a uniform electric field ( E

), net force on the dipole is zero

and it experiences a torque where p E p qd

.

(b) In non-uniform field, dipoles have net force F p E

and torque Ep .

(c) Energy of an ideal dipole p in an electric field E

is .U p E

.

(d) Interaction energy of two dipoles separated by a distance r is

rprpppr

U ˆˆ34

121213

0

.

1.8.2 Approximate potential at large distances

Approximate potential at large distances due to arbitrary localized charge distribution

2' ' ' ' ' ' ' 2 ' ' '2 2

0

1 1 1 1 3 1cos cos ...4 2 2

V r r d r r d r r dr r r

r

y

z

x

p

PR

r'r

'd





The first term (n = 0) is the monopole contribution (it goes like r1

). The second term

(n = 1) is the dipole term (it goes like 2

1r ). The third term is quadrupole; the fourth

octopole and so on.

The lowest nonzero term in the expansion provides the approximate potential at large r

and the successive terms tell us how to improve the approximation if greater precision is

required.

The monopole and dipole terms

Ordinarily, the multipole expansion is dominated (at large r) by the monopole term:

0

14mon

QV rr

.

where dQ is the total charge of the configuration.

If the total charge is zero, the dominant term in the potential will be the dipole (unless, of

course, it also vanishes):

' ' ' ' ' ' '2 2 2

0 0 0

ˆ1 1 1 1 1 .ˆcos .4 4 4dip

r pV r r r d r r r dr r r

,

where dipole moment ' ' 'p r r d

.

The dipole moment is determined by the geometry (size, shape and density) of the charge

distribute. The dipole moment of a collection of point charge is

'

1

n

i ii

p q r

.

Note: Ordinarily, the dipole moment does change when we shift the origin, but there is

an important exception: If the total charge is zero, then the dipole moment is independent

of the choice of origin.





1.9 Polarization

(Polarization) P

dipole moment per unit volume

1.9.1 The Field of a polarized Object (Bound Charges)

Suppose we have a piece of polarized material with

polarization vector P

containing a lot of microscopic

dipoles lined up. Then

0 0

1 1 1 1. ' '. '4 4S V

V r P d a P dR R

.

The first term looks like the potential of a surface bound charge ˆ.b P n

(where n̂ is

the normal unit vector).

The second term looks like the potential of a volume bound charge .b P

.

Thus 0 0

1 1' '4 4

b b

S V

V r da dR R

, this means the potential (and hence also the

field) of a polarized object is the same as that produced by a volume charge

density .b P

plus a surface charge density ˆ.b P n

.

1.10 The Electric Displacement

1.10.1 Gauss Law in the Presence of Dielectrics

Within the dielectric, the total charge density can be written as b f where b is

volume bound charge f free charge density.

0. fD where D E P

is known as the electric displacement.

Thus Gauss’ law reads, . fD

Or, in integral form .encfD da Q

, where

encfQ denotes the total free charge enclosed

in the volume.

p

R





1.10.2 Linear Dielectrics (Susceptibility, Permittivity, Dielectric Constant)

For any substances, the polarization is proportional to the field provided

is not too

strong: 0 eP E P E

(Materials that obey this relation are called linear dielectrics)

The constant of proportionality, e is called the electric susceptibility of the medium.

The value of e depends on the microscopic structure of the substance and also on

external conditions such as temperature.

In linear media we have

0 0 0 0 0(1 ) , (1 )e e eD E P E E E E where

This new constant is called the permittivity of the material.

Also0

(1 )r e

is called relative permittivity or dielectric constant, of the material.

1.10.3 Boundary Condition on D

The boundary between two medium is a thin sheet of free surface charge f .

fbelowabove DD and ||||||||belowabovebelowabove PPDD

1.10.4 Energy in dielectric system

12 all space

W D E d

.





1.12 Image problems

1.12.1 The Classic Image problem

Suppose a point charge q is held a distance d above an infinite grounded conducting

plane. We can find out what is the potential in the region above the plane.

Forget about the actual problem; we are going to study a complete different situation.

The new problem consists of two point charges +q at (0, 0, d) and –q at (0, 0,-d) and no

conducting plane.

For this configuration we can easily write down the potential:

2 22 2 2 20

1 q qV x, y, z4 x y z d x y z d

(The denominators represent the distances from (x, y, z) to the charges +q and –q,

respectively.) It follows that

1. 0V when 0z and

2. 0V for 2 2 2 2x y z d ,

and the only charge in the region 0z is the point charge +q at (0, 0, d). Thus the

second configuration produces exactly the same potential as the first configuration, in the

upper region 0z .

q

z

dy

x

dq

0V

qz

dy

x





1.12.2 Induced Surface Charge

The surface charge density induced on the conductor surface is 00z

Vz

32 2 2 2

,2

qdx yx y d

As expected, the induced charge is negative (assuming q is positive) and greatest at

x=y=0.

The total induced charge Q da q

1.12.3 Force and Energy

The charge q is attracted towards the plane, because of the negative induced surface

charge. The force:

2

20

1 ˆ4 2

qF zd

.

One can determine the energy by calculating the work required to bring q in from

infinity.

2 2 2

20 0 0

1 1 1.4 4 4 4 4 4

dd d q q qW F dl dzz z d

1.12.4 Other Image Problem

The method just described is not limited to a single point charge; any stationary charge

distribution near a grounded conducting plane can be treated in the same way, by

introducing its mirror image.

r'R

R

q'qb

a

qaR

0V





Let us examine the completely different configuration, consisting of the point charge q

together with another point chargeRq ' qa

placed at a distance2Rb

a to the right of

the centre of sphere. No conductor, now-just two point charges.

The potential of this configuration is 0

1 q q 'V(r, )4 R R

2 2 2 20

q 1 1V(r, )4 r a 2ra cos R (ra / R) 2ra cos

Clearly when r = R, V 0

Force

The force on q, due to the sphere, is the same as the force of the image charge q, thus:

2

2 2 20

1 q RaF4 (a R )

Energy

To bring q in from infinity to a, we do work

2

2 20

1 q RW4 2(a R )





2. MAGNETOSTATICS

The magnetic field at any point due to steady current is called as magnetostatic field.

2.1 Magnetic force on current element

The magnetic force on a charge Q, moving with velocity v in a magnetic field B

is,

BvQF mag . This is known as Lorentz force law. In the presence of both electric and

magnetic fields, the net force on Q would be: BvEQF

2.1.1 Current in a wire A line charge λ traveling down a wire at a

speed v constitutes a current I v

.

Magnetic force on a segment of current-

carrying wire is, magF I dl B

2.1.2 Surface current density

When charge flows over a surface, we describe it by the surface current K

.

dldK is the current per unit width-

perpendicular to flow.

Also K v

where is surface charge

density and v

is its velocity.

Magnetic force on surface current magF K B da

2.1.3 Volume current density

dJda

is the current per unit area-perpendicular to

flow. Also J v

where is volume charge density

and v

is its velocity. Magnetic force on volume

current magF J B d

.Current crossing a surface S is S

J d a

P

tvv

K

Flowdl

J

da

Flow





2.2 Continuity equation

General continuity equation t

J

This is the precise mathematical statements of local charge conservation.

Thus for magnetostatic fields 0

t

and hence the continuity equation becomes:

0J

.

2.3 Biot-Savart law (magnetic field of steady line current) The magnetic field of a steady line current is given by

0 02 2

ˆ ˆ'4 4

I R dl RB r dl IR R

70 24 10 ( )Nwhere permeability of free space

A

For surface and volume current Biot - Savart law becomes:

0 02 2

ˆ ˆ( ') ( ')( ) ' ( ) '4 4

K r R J r Rr da and r dR R

.

2.3.1 Magnetic field due to wire

Let us find the magnetic field a distance d from a long straight wire carrying a steady

current I.

02 1

I ˆB (sin sin )4 d

For Infinite wire: 1 2

and 2 2

0I ˆB2 d

Note:

1. Force (per unit length) of attraction between two

long, parallel wires a distance d apart, carrying

currents I1 and I2 in same direction are: dIIf 210

2

.

2. If currents are in opposite direction they will repel with same magnitude.

I

R

'dI

r

1

I

2

Wire segment





3. The magnetic field a distance d above the center of a circular loop of radius R, which

carries a steady current I is2

02 2 3/2

I R ˆB z2 (R d )

.

At the center of the circle 0I ˆB(0) z2R

2.3.2 Magnetic field due to Solenoid and Toroid

The magnetic field of a very long solenoid, consisting of n closely wound turns per unit

length of a cylinder of radius R and carrying a steady current I is:

0 ˆ inside the solenoid0 outside the solenoid.

nI zB

Magnetic field due to toroid is^

0 for points inside the coil 2

0 for points outside the coil

NIB r

where N is the total number of turns.

2.4 Ampere's Law

In general we can write 0 encB dl I

where .encI J d a

is the total current enclosed

by the amperian loop.

0B J

Right hand Rule

If the fingers of your right hand indicate the direction of integration around the boundary,

then your thumb defines the direction of a positive current.

2.5 Magnetic Vector Potential A : Since 0B B A

For magnetostatic fields, 200A and A J

If J goes to zero at infinity, 0 '' .

4J r

A r d for volume currentR

For line and surface currents, 0 01 '; '4 4

I KA r dl A r daR R





2.6 Magnetostatic boundary condition (Boundary is sheet of current, K

)

Just as the electric field suffers a discontinuity at a surface charge, so the magnetic field

is discontinuous at a surface current. Only this time it is the tangential component that

changes.

Since . 0B d a

above belowB B

For tangential components

|| ||0 0

|| ||0

. is parallel to surface but to

. is parallel to surface and along

renc above below

enc above below

B dl I B B K B K

B dl I B B B K

Thus the component of B

that is parallel to the surface but perpendicular to the current

is discontinuous in the amount 0K . A similar amperian loop running parallel to the

current reveals that the parallel component is continuous. The result can be summarized

in a single formula:

nKBB belowabove ˆ0 ,

where n̂ is a unit vector perpendicular to the surface, pointing “upward”.

Like the scalar potential in electrostatics, the vector potential is continuous across, an

boundary: above belowA A

For . 0A

guarantees that the normal component is continuous, and A B

, in the

form . .line S

A dl B d a

But the derivative of A

inherits the discontinuity of B

: 0above belowA A Kn n





2.7 Multiple Expansion of Vector Potential

We can always write the potential in the form of a power series in 1r

, where r is the

distance to the point in question. Thus we can always write

.........'

21'cos

23'1''cos'1'1

422

320 ldr

rldr

rld

rIrA

First term, monopole 0dl

(no magnetic

monopole)

Second term, dipole 02

ˆ4

dipm rA r

r

where m is the magnetic dipole moment:

AIadIm where A is area vector

Thus

ˆsin4 2

0

rmrAdip

Hence

ˆsinˆcos24 3

0 rrmArB dip 0

3

1 ˆ ˆ34

m r r mr

Note:

(a) When a magnetic dipole is placed in a uniform magnetic field ( B

), net force on the

dipole is zero and it experiences a torque m B

.

(b) In non-uniform field, dipoles have net force F m B

and torque Bm .

(c) Energy of an ideal dipole m in an magnetic field B

is .U m B

.

(d) Interaction energy of two dipoles separated by a distance r is

rmrmmmr

U ˆˆ34

121213

0

.

.

m

x

r

z

y





2.8 Magnetisation M

Magnetization M is magnetic dipole moment per unit volume.

2.8.1 The Field of a magnetized Object (Bound Currents)

Consider a piece of magnetized material with magnetization M .

Then the vector potential is given by

0 01 1'4 4v

A r M r d M r daR R

.

This means the potential(and hence also the field) of a magnetized object is the same as

would be produced by a volume current MJb throughout the material, plus a

surface current nMKb ˆ , on the boundary. We first determine these bound currents,

and then find the field they produce.

2.9 The Auxiliary field ( H

)

2.9.1 Ampere’s Law in in presence of magnetic materials

In a magnetized material the total current can be written as b fJ J J

where bJ

is

bound current and fJ

is free current. Thus fH J

where0

BH M

In integral formencfH dl I

where

encfI is the total free current passing through the

amperian loop.

H

plays a role in magnetostatic analogous to D

in electrostatic: Just as D

allowed us to

write Gauss's law in terms of the free charge alone, H

permits us to express Ampere's

law in terms of the free current alone- and free current is what we control directly.

Note:

When we have to find B

or H

in a problem involving magnetic materials, first look for

symmetry. If the problem exhibits cylindrical, plane, solenoid, or toroidal symmetry, then

we can get H

directly from the equationencfH dl I

.





2.9.2 Magnetic Susceptibility and Permeability

For most substances magnetization is proportional to the field H , HM m ,

where m is magnetic susceptibility of the material.

HBHMHB m 100 where mr 1000 , is

permeability of material.

2.9.3 Boundary Condition ( H

)

The boundary between two medium is a thin sheet of free surface current fK .

The Ampere’s law states that

ˆabove below fH H K n

.

And belowabovebelowabove MMHH





3. Dynamics of charged particles in static and uniform electromagnetic fields

(The Lorentz Force Law)


is,

BvQF mag

This is known as Lorentz force law.

In the presence of both electric and magnetic fields, the net force on Q would be:

F Q E v B

3.1 Charged particle in static electric field

3.1.1 Charged particle enters in the direction of field (Linear motion)

The force on the charge Q in electric field E is EQF .

Acceleration of the charge particle in the direction of the electric field is mEQ

mFa .

If r is the position vector at any time t then

Let at t=0, 0rr 01 rC 02

2rtut

mEQr

If initially 0,0 0 ru 2

2t

mEQr and t

mEQv .

The energy acquired by the charged particle in moving from point 1 to 2 is

2

1

2

1

2

1

2

1

.... vdvmdtvdtvdmldamldFW 2

1222

1 vvmW

If the potential difference between points 1 to 2 is V then 21

222

1 vvmQVW

If the particle starts from rest i.e 01 v and final velocity is v then

QE

1 2

ut

1 2

0r 2

2QE t

m





mQVvmvQVW 2

21 2

Kinetic energy of the particle QErtm

QEQEtm

EQmmvEK 222

222

2.

21

21..

3.1.2 Charged particle enters in the direction perpendicular to field (Parabolic

motion)

Let us consider a charge particle enters in an electric field region with velocity xv at t=0.

The electric field is in the y-direction and the field region has length l. After traversing a

distance l it strikes a point P on a screen which is placed at a distance L from the field

region.

Thus 2

2

221

x

yy v

xm

QEtay and which represents parabolic path.

2

1 2

x

y

vl

mQE

y and tan2 Ly

Thus distance of point P from the center of the screen is, tan2

2

21 Lvl

mQE

yyx

y

Angle of deviation in the field region, xmvQE

dxdy

x

y2tan

Angle of deviation in the field free region, lmvQE

x

y2tan

L

xvyE

l

1y

2y

x

yP





3.2 Charged particle in static magnetic field


is,

BvQF mag

This is known as Lorentz force law.

3.2.1 Charged particle enters in the direction perpendicular to field (Circular

motion)

If a charge particle enters in a magnetic field at angle of 90o, then motion will be circular

with the magnetic force providing the centripetal acceleration.

For uniform circular motion: 2v mvQvB m R

R QB

where R is the radius of the circle and m is the mass of the charge

particle.

Momentum of the charged particle p QBR

Kinetic energy (KE) m

RBQm

p22

2222

Time periodQB

mvRT 22

3.2.2 Charged particle enters in the direction making an angle with the field

(Helical motion)

If the charge particle enters in a magnetic field making an angle θ,

then motion will be helical.

||sin and cosv v v v ,

and the radius of helix is mvRQB

.

F

Bz

Qx

v

y

R

B||





3.3 Charged particle in uniform electric and magnetic field (Cycloid motion)

ˆˆ ˆ

ˆ ˆ

,

( )

F Q E v B Q E z Bz y Byz

ma m y y zz

Ey z z yB

QBwhere cylotron frequencym

Solving above differential equations, we get

2 2 2

sin , 1 cosE Ey t t t z t tB B

Ey R t z R R where RB

This is the formula for a circle, of radius R,

whose center is 0, ,R t R travels in the y-direction at constant speed, BERv

The curve generated in this way is called a cycloid.

Magnetic forces do not work because Bv is perpendicular to v , so

0 dtvBvQldFdW magmag .

Magnetic forces may alter the direction in which a particle moves, but they can not speed

up or slow down it.

E

o

B

x

z

ya b c





4. Electromagnetic induction

4.1 Faraday’s Law

Experiment 1: He pulled a loop of wire to the right through a magnetic field. A current

flowed in the loop (Figure a).

Experiment 2: He moved the magnet to the left, holding the loop still. Again, a current

flowed in the loop (Figure b).

Experiment 3: With both the loop and the magnet at rest, he changed the strength of the

field (he used an electromagnet, and varied the current in the coil). Once again current

flowed in the loop (Figure c).

Thus, universal flux rule is that, whenever (and for whatever reason) the magnetic flux

through a loop changes, an e.m.f. will appear in the loop.

ddt

(Where magnetic flux .B da

)

In experiment 2, A changing magnetic field induces an electric field.

It is this “induced” electric field that accounts for the e.m.f. dE dldt

Then .BE dl dat

tBE

4.1.1 Lenz’s Law

In Faraday’s law negative sign represents the Lenz’s law. (The induced current will flow

in such a direction that the flux it produces tends to cancel the change).

For example if the magnetic flux is increasing then induced e.m.f will try to reduce and

vice versa.

c

I

B

field magnetic changing

vI

B

a

vI

B

b





5. Maxwell's equations

5.1 Maxwell’s equation in free space

5.1.1 Electrodynamics before Maxwell’s

(i) 0

(Gauss’ Law),

(ii) 0B

(No name),

(iii) Bt

(Farday’s Law),

(iv) 0B J

(Ampere’s law).

5.1.2 How Maxwell fixed Ampere’s Law

From continuity equation and Gauss Law

0 0. ( . ) . EJ Et t t

0. 0EJt

.

Thus 0 0 0EB Jt

A changing electric field induces a magnetic field.

Maxwell called this extra term the displacement current 0dEJt

.

Integral form of Ampere's law 0 0 0. .encEB dl I dat

5.1.4 Maxwell’s equation in free space

(i) 0

(Gauss’ Law),

(ii) 0B

(No name),

(iii) Bt

(Farday’s Law),

(iv) 0 0 0EB Jt

(Ampere’s law with Maxwell's correction).





5.3 Boundary conditions on the fields at interfaces

( )

( )above below f

above below

a D D

b B B

|| ||

|| ||

( )

ˆ( )

above below

above below f

c E E

d H H K n

In particular, if there is no free charge or free current at the interface between medium1

and medium 2, then

1 1 2 2

1 2

( ) 0

( )

a E E

b B B

and

|| ||1 2

|| ||1 2

1 2

( )1 1( ) 0

c E E

d B B

.

6. Electromagnetic waves in free space

6.1 Poynting Theorem (“work energy theorem of electrodynamics”)

The work necessary to assemble a static charge distribution is

20

2eW E d , where E is the resulting electric field

The work required to get currents going (against the back emf) is

2

0

12mW B d

, where B is the resulting magnetic field

This suggests that the total energy in the electromagnetic field is

2

20

0

12em

BU E d

.

Suppose we have some charge and current configuration which at time t, produces

fields &E B

. In next instant, dt, the charges moves around a bit. The work is done by

electromagnetic forces acting on these charges in the interval dt.

According to Lorentz Force Law, the work done on a charge ‘q’ is

( )F dl q E v B v dt qE v dt

.





Now q d and v J

, so the rate at which work is done on all the charges in a

volume V is

2 20

0 0

1 1 12V S

dW d E B d E B d adt dt

,

where S is the surface bounding V.

This is Poynting's theorem; it is the “work energy theorem” of electrodynamics.

The first integral on the right is the total energy stored in the fields, emU .

The second term evidently, represents the rate at which energy is carries out of V, across

its boundary surface, by the electromagnetic fields.

Poynting's theorem says, that, “the work done on the charges by the electromagnetic

force is equal to the decrease in energy stored in the field, less the energy that flowed out

through the surface”.

The energy per unit time, per unit area, transported by the fields is called the Poynting

vector

0

1S E B

.

adS is the energy per unit time crossing the infinitesimal surface d a

– the energy or

energy flux density.

Momentum density stored in the electromagnetic field is: 0 0dem S

6.2 Waves in one dimension (Sinusoidal waves)

6.2.1 The wave equation

A wave propagating with speed v in z-direction can be expressed as: 2 2

2 2 2

1f fz v t

The most general solution to the wave equation is the sum of a wave to the right

(+z direction) and a wave to the left (-z direction):

,f z t g z vt h z vt .





6.2.2 Terminology

Let us consider a function , cos ( )f z t A k z vt

A is the amplitude of the wave (it is positive, and represents the maximum displacement

from equilibrium).

The argument of the cosine is called the phase, and is the phase constant (normally,

we use a value in the range 0 2 ).

Figure given below shows this function at time 0t . Notice that at z vt k , the phase

is zero; let's call this the “central maximum.” If 0 , central maximum passes the origin

at time 0t ; more generally k is the distance by which the central maximum (and

therefore the entire wave) is “delayed.”

Finally k is the wave number; it is related to the wavelength as 2k

, for when z

advances by 2k , the cosine executes one complete cycle.

As time passes, the entire wave train proceeds to the right, at speed v . Time period of

one complete cycle is 2Tkv

.

The frequency (number of oscillations per unit time) is 12kv v

T

.

The angular frequency 2 kv

In terms of angular frequency , the sinusoidal wave can be represented as

, cosf z t A kz t .

A

/ k z

, 0f zCentral

maximum

v





A sinusoidal oscillation of wave number k and angular frequency traveling to the left

would be written

, cosf z t A kz t .

Comparing this with the wave traveling to the right reveals that, in effect, we could

simply switch the sign of k to produce a wave with the same amplitude, phase constant,

frequency, and wavelength, traveling in the opposite direction.

6.2.3 Complex notation

In view of Euler's formula,

cos sinie i ,

the sinusoidal wave , cosf z t A kz t can be written as

, Re i kz tf z t Ae ,

where Re denotes the real part of the complex number . This invites us to introduce

the complex wave function

, i kz tf z t Ae

with the complex amplitude iA Ae absorbing the phase constant.

The actual wave function is the real part of f :

, Re ,f z t f z t .

The advantage of the complex notation is that exponentials are much easier to manipulate

than sines and cosines.

/ kz

, 0f zCentral

maximum

v





6.2.4 Polarization

In longitudinal wave, the displacement from the equilibrium is along the direction of

propagation. Sound waves, which are nothing but compression waves in air, are

longitudinal.

Electromagnetic waves are transverse in nature. In a transverse wave displacement is

perpendicular to the direction of propagation.

There are two dimensions perpendicular to any given line of propagation. Accordingly,

transverse waves occur in two independent state of polarization:

“Vertical” polarization ˆ, i kz tvf z t Ae x ,

“Horizontal” polarization ˆ, i kz thf z t Ae y ,

or along any other direction in the xy plane ˆ, i kz tf z t Ae n

The polarization vector n̂ defines the plane of vibration. Because the waves are

transverse, n̂ is perpendicular to the direction of propagation:

ˆ ˆ. 0n z

6.3 Electromagnetic waves in vacuum

6.3.1 The wave equation for E and B

Write Maxwell’s equations in free space ( 0 0and J

) then,

2

20 0 2

EEt

and 2

20 0 2

BBt

Thus, E and B satisfy the wave equation 2

22 1

tf

vf

.

So, EM waves travels with a speed

8

0 0

1 3 10 / ( )v m s c velocity of light in free space

27 122 20 04 10 , 8.86 10N Cwhere A Nm





6.3.2 Monochromatic plane waves

Suppose waves are traveling in the z-direction and have no x or y dependence; these are

called plane waves because the fields

are uniform over every plane

perpendicular to the direction of

propagation.

The plane waves can be represented as:

tkzieEtzE 0~,~ , tkzieBtzB 0

~,~

where 0~E and 0

~B are the (complex)

amplitudes (the physical fields, of course are the real parts of E~ and B~ ).

That is, electromagnetic waves are transverse: the electric and magnetic fields are

perpendicular to the direction of propagation.

Also 0 0ˆ( )kB z E

There is nothing special about the z direction; we can generalize the monochromatic

plane waves traveling in an arbitrary direction. The propagation vector or wave vector k

points in the direction of propagation, whose magnitude is the wave number k. The scalar

product .k r

is the appropriate generalization of kz, so

. .0 0

1 1ˆ ˆˆ ˆ, , ,i k r t i k r tE r t E e n B r t E e k n k Ec c

where n̂ is polarization vector.

The actual (real) electric and magnetic fields in a monochromatic plane wave with

propagation vector k

and polarization n̂ are

0 01 ˆˆ ˆ, cos . , , cos .E r t E k r t n B r t E k r t k nc

c0

c/0

z

y

x





6.3.3 Energy and Momentum in Electromagnetic Wave

The energy per unit volume stored in electromagnetic field is 2 20

0

1 12

u E B

In case of monochromatic plane wave 2

2 20 02

EB Ec

So the electric and magnetic contributions are equal i.e. 2 20

0

1 12 2E Bu u E B

.

2 2 20 0 0 = cos ( )E Bu u u E E kz wt .

As the wave travels, it carries this energy along with it. The energy flux density (energy

per unit area, per unit time) transported by the fields is given by the Pointing vector

0

1 ( )S E B

For monochromatic plane wave propagating in the z-direction,

2 20 0 ˆ ˆcos ( )S c E kz wt z cuz

.

The energy per unit time, per unit area, transported by the wave is therefore uc.

Electromagnetic fields not only carry energy, they also carry momentum. The

momentum density stored in the field is 2

1 Sc

.

For monochromatic plane wave, 2 20 0

1 1ˆ ˆcos ( )E kz wt z uzc c

.

Average energy density 20 0

12

u E ,

Average of Poynting vector ^

20 0

12

S c E z

,

Average momentum density ^

20 0

12

E zc

.

The average power per unit area transported by an electromagnetic wave is called the

intensity 20 0

12

I S c E .





Note:

(a) When light falls on perfect absorber it delivers its momentum to the surface. In a

time t the momentum transfer is p Ac t ,

so the radiation pressure (average force per unit area) is

20 0

1 12

p IP EA t c

.

(b) When light falls on perfect reflector, the radiation pressure

cIP 2

because the momentum changes direction, instead of being absorbed.

6.4 Electromagnetic waves in matter

Inside matter, but in regions where there is no free charge or free current. If the medium

is linear and homogeneous, 1 and H=D E B

.

Now the wave equation inside matter is2 2

2 22 2

E BE and Bt t

.

Thus EM waves propagate through a linear homogenous medium at a speed

1 cvn

where 0 0

n

thus rn is the index of refraction (since 1r for non-magnetic material).

The energy density 2 21 12

u E B

20

12

u E

The Poynting vector BES

1 ^

20

12

S v E z

Intensity 20

12

I S vE

Thus in a medium c→ v, 0 and 0

c

tc

A





6.5 Electromagnetic waves in conductors

Any initial free charge density 0f given to conductor dissipate in a characteristic time

/ where σ is conductivity and / 0tf ft e .

This reflects the familiar fact that if we put some free charge on conductor, it will flow

out to the edges. The modified wave equation for E and B

are,

2 2

2 22

E E B BE and Bt t t t

The admissible plane wave solution is tzkieEtzE ~

0~,~ , tzkieBtzB

~0

~,~

“wave number” k is complex. Let k k i where k and are real and imaginary part

of k . 1/2 1/2

2 2

1 1 12 2

k and

Thus, 0 0, , ,i kz t i kz tz zE z t E e e B z t B e e

The distance it takes to reduce the amplitude by a factor of e1 is called the skin depth (d)

1d

;

it is a measure of how far the wave penetrates into the conductor.

The real part of k determines the wavelength, the propagation speed, and the index of

refraction: k 2

, vk

, ckn

Like any complex number, k can be expressed in terms of its modulus and phase:

ik Ke

where 2

2 2 1K k k

and 1tan k





The complex amplitudes 0 0 0 0E Bi iE E e and B B e are related by

0 0 0 0B E

ii ik KeB E B e E e

.

Evidently the electric and magnetic fields are no longer in phase; in fact B E , the

magnetic field lags behind the electric fields.

1/22

0

0

1B KE

Thus,

0

0

ˆ, cos

ˆ, cos

zE

zE

E z t E e kz t x

B z t B e kz t y

Note:

(a) In a poor conductor ( )

2

i.e independent of frequency.

(b) In a very good conductor ( )

1 2 1,2

df

(c) When an electromagnetic wave strikes a perfect conductor ( ) then all waves

are reflected back i.e. 0 0 0and 0R I TE E E .





7. Applications of Electromagnetic waves

7.1 Reflection and Refraction

7.1.1 Normal incidence

At 0z , the combined field on the left I R I RE E and B B , must join the fields on the right

&T TE B , in accordance with the boundary conditions.

2 1 20 0 0 0

2 1 1 2

2 , R I T Iv v vv v v v

Note:

Reflected wave is in phase if 2 1 2 1v v or n n

and out of phase if 2 1 2 1v v or n n .

In terms of indices of refraction the real

amplitudes are

1 2 10 0 0 0

1 2 1 2

2 , R I T In n nn n n n

.

Since Intensity 20

12

I vE , then the ratio of the reflected intensity to the incident intensity is

the Reflection coefficient 2 2

0 1 2

0 1 2

RR

I I

EI n nRI E n n

.

The ratio of the transmitted intensity to the incident intensity is the Transmission coefficient

2

02 2 1 22

1 1 0 1 2

4TT

I I

EI v n nTI v E n n

.

1R T

1 2

zy

x

Interface

I

I

1v

T

2vT

R1v

R





7.1.2 Oblique incidence

In oblique incidence an incoming wave meets the boundary at an arbitrary angle I . Of course,

normal incidence is really just a special case of oblique incidence with 0I .

First Law (law of refraction)

The incident, reflected and transmitted wave vectors

form a plane (called the incidence plane),

which also includes normal to the surface.

Second law (law of reflection)

The angle of incidence is equal to the angle of reflection

i.e. I R

Third Law: (law of refraction, or Snell’s law)

1

2

sinsin

T

I

nn

.

7.1.3 Fresnel’s relation (Parallel and Perpendicular Polarization)

Case I: (Polarization in the plane of incidence)

Reflected and transmitted amplitudes

0 0R IE E

and 0 0

2T IE E

where 1 1 1 2

2 2 2 1

coscos

T

I

v nandv n

These are known as Fresnel’s equations.

Notice that transmitted wave is always in phase with

the incident one; the reflected wave is either in phase ,

if , or 1800 out phase if .

At Brewster’s angle ( B ) reflected light is completely extinguished when , or

1

2tannn

B 90oI Band

Tk

Ik

Rk

2

Tz

R

I Plane of Incidence

1

RB

RE

Ik

Tk

Rk

2

Tz

R

I

1I

I

T

T





When light enters from denser to rarer medium ( 21 nn ) then after a critical angle ( C )

there is total internal reflection.

0

1 2

2 1

sin 90 sinsin c

c

n nn n

, 90oC Tat .

Reflection and Transmission coefficients are 2

R and

22

T

1 TR

CaseII: (Polarization perpendicular to plane of incidence)

ITIR EEandEE 0000 12

11

In this case B is not possible.

Thus 2 2

1 21 1

R and T

1R T

Ik

Tk

Rk

2

Tz

R

I

1

RB

RE

I

I

T

T





8. Potential and Field formulation for Time Varying Fields

8.1 Scalar and vector potentials

Now from first Maxwell’s equation (i)

2

0

V At

...............(1)

From fourth Maxwell’s equation

22

0 0 0 0 02

A VA A Jt t

…............(2)

Equations (1) and (2) contain all the information of Maxwell’s equations. Thus we need

to calculate only four components (one for V and three for A

) instead of six components

(three for E

and three for B

).

8.2 Gauge transformation

Suppose we have two sets of potentials, ,V A

and ,V A

, which correspond to the

same electric and magnetic fields.

Thus A A

andV V .

It follows that ,

.

A A

V Vt

Conclusion: For any old scalar function , we can add

to A

, provided we

simultaneously subtract t

from V. None of these will affect the physical quantities E

and B

. Such changes inV and A

are called gauge transformations.

B A

AE Vt





8.3 Coulomb and Lorentz gauge

Coulomb Gauge reads 0A

.

Lorentz Gauge condition is 0 0VAt

.

Since22

0 0 0 0 02A VA A J

t t

and 2

0

V At

,

Using Lorentz Gauge condition

22

0 0 02

AA Jt

and22

0 0 20

VVt

.

The virtue of the Lorentz gauge is that it treatsV and A

on an equal footing: the same

differential operator 22 2

0 0 2t

(called the d' Alembertian) occurs in both

equations:

(i) 2

0

V

, (ii) 2oA J

.

9.2 Waveguides

Electromagnetic waves confined to the interior of a hollow pipe or wave guide. The wave

guide is a perfect conductor, 00 BandE

inside the material itself, and hence the

boundary conditions at the inner wall are:

||

0E

and 0B

Free charges and currents will be induced on the surface in such a way as to enforce these

constraints. Let us assume E.M. Waves that propagate inside the waveguide is

represented by:

0 0, , , ( , ) , , , , ( , )i kz t i kz tE x y z t E x y e B x y z t B x y e

.





These electric and magnetic field must satisfies Maxwell's equations in the interior of the

waveguide.Since confined waves are not (in general) transverse; in order to fit the

boundary conditions we shall have to include longitudinal components z zE and B :

0 0ˆ ˆ ˆ ˆˆ ˆ,x y z x y zE E x E y E z B B x B y B z .

Putting this into Maxwells equations(iii) and (iv) and compare R.H.S and L.H.S

zy x

E ikE i By

, zx y

EikE i Bx

, y xz

E E i Bx y

2z

y xB iikB Ey c

, 2

zx y

B iikB Ex c

, 2y x

z

B B i Ex y c

Let us rewrite these six equations

(i) y xz

E E i Bx y

, (iv) 2y x

z

B B i Ex y c

(ii) zy x

E ikE i By

, (v) 2z

y xB iikB Ey c

(iii) zx y

EikE i Bx

, (vi) 2z

x yB iikB Ex c

Equation (ii), (iii), (v), and (vi) can be solved for xE , yE , xB , and yB :

(i) 2

2

z zx

E BiE kx ykc

(ii) 2

2

z zy

E BiE ky xkc

(iii) 2 2

2

z zx

B EiB kx c ykc

(iv) 2 2

2

z zy

B EiB ky c xkc





Finally, we will get coupled differential equation

(i) 22 2

22 2 0zk Ex y c

(ii)

22 22

2 2 0zk Bx y c

If 0zE we call these TE (“transverse electric”) waves; if 0zB they are called TM

(“transverse magnetic”) waves; if both 0zE and 0zB , we call them TEM waves.

Note: It turns out that TEM waves can not occur in a hollow waveguide.

9.2.1 TE Waves in Rectangular Waveguide

Suppose we have a wave guide of rectangular shape with height a and with b a b , and

we are interested in the propagation of TE waves.

The problem is to solve 22 2

22 2 0zk Bx y c

(since 0zE )

subject to boundary condition 0B

.

Thus

byn

axmBBz

coscos0 where ......2,1,0.....2,1,0 nandm

x

a

b

y

z





This solution is called the mnTE mode. (The first index is conventionally associated with

the larger dimension.)

Wave number k is obtaine from equation 2

2 2 2 0x xk k kc

by putting xk and yk .

So wave number

2222

bn

am

ck where f 2 .

If mnbn

amc

22 or

22

2 bn

amcfmn ,

the wave number is imaginary, and instead of a traveling wave we have exponentially

attenuated fields. For this reason mn or mnf is called cutoff frequency for the mode in

question.

The lowest cutoff frequency (fundamental mode) for the waveguide occurs for the

mode 10TE :

a

c 10 or acf

210 .

Frequencies less than this will not propagate at all.

Wave number can be written more simply in terms of the cutoff frequency

221mnc

k .

The wave velocity is 2 2

1 1mn mn

c cv ck f

f

.

Group velocity 2 21 1 1mn mn

gfv c c cfdk

d

.





Wavelength inside the waveguide 0 02 2

1 1g

mn mnff

Characteristic impedance2 2

377 377

1 1TE

mn mnff

9.2.2 TM Waves in Rectangular Waveguide

The problem is to solve 22 2

22 2 0zk Ex y c

(since 0zB )

subject to boundary condition||

0E

.

We will get similar expression as we have derived in TE waves. Thus

byn

axmEEz

sinsin0 where ......3,2,1.....3,2,1 nandm

This solution is called the mnTM mode. (The first index is conventionally associated

with the larger dimension.)

Formula for cutoff frequency mn , wave velocityv , group velocity gv and g are same as

TE waves.

Characteristic impedance 2

1377

w

wmnTM

The fundamental mode is 11TM and 2 2

111 1

2cf a b .





10. Radiation- from moving charges

10.1 Radiation

Accelerating charges and changing current produce electromagnetic waves (radiation).

If the source is localized near the origin, the total power passing out through the spherical

shell of radius r is the integral of pointing vector:

0

1. .P r S da E B da

The power radiated is the limit of this quantity as r goes to infinity:

limrad rP P r

This is the energy (per unit time) that is transported out to infinity, and never comes back.

10.1.1 Power radiated by a point charge

The electric field E

and magnetic field B of an EM wave due to a point charge q, having

acceleration a

at any point P (position vector r

) is given by

rqaE sin

, rqaB sin

(where is the angle which r makes with z-direction)

Thus rrc

aqr

aqS ˆsin16

sin2

2

2

220

2

222

The total power radiated is evidently

caqddr

rcaqadSP

6

sinsin16

.22

022

2

2

220

The direction of E

can be determined from the following rule:

(i) .r and E is always perpendicular

(ii) Eandra, lies in one plane.

r

Source

z

y

x

Pr

a

E

q

emt formula sheet bsc level

Documents