EEE PPT

• NAME:HARSHIL.R.SHAH• EN NO:151080106025• BRANCH:CIVIL• SEM:2• TOPIC;OHM’S LAW,GAUSS LAW,

FARADE’S LAW• SUMBITED BY• ANKUR SIR

Ohm’s Law

Every conversion of energy from one form to another can be related to this equation.

In electric circuits the effect we are trying to establish is the flow of charge, or current. The potential difference, or voltage between two points is the cause (“pressure”), and resistance is the opposition encountered.

OppositionCause

Effect

Ohm’s Law

Simple analogy: Water in a hose Electrons in a copper wire are analogous to water in

a hose. Consider the pressure valve as the applied voltage

and the size of the hose as the source of resistance. The absence of pressure in the hose, or voltage across

the wire will result in a system without motion or reaction. A small diameter hose will limit the rate at which water will

flow, just as a small diameter copper wire limits the flow of electrons.

Ohm’s Law

Developed in 1827 by Georg Simon Ohm For a fixed resistance, the greater the voltage

(or pressure) across a resistor, the more the current.

The more the resistance for the same voltage, the less the current.

Current is proportional to the applied voltage and inversely proportional to the resistance.

Ohm’s Law

Where: I = current (amperes, A)E = voltage (volts,

V)R = resistance

(ohms, )

RE

I

4.3 - Plotting Ohm’s Law

Plotting Ohm’s Law

•Insert Fig Insert Fig 4.84.8

2). Gauss’ Law and Applications• Coulomb’s Law: force on charge i due to

charge j is

• Fij is force on i due to presence of j and acts along line of centres rij. If qi qj are same sign then repulsive force is in direction shown

• Inverse square law of force

ˆ

ˆ

ji

jiijjiijjiij

ij2ij

ji

oji3

ji

ji

oij

r

rqq

41qq

41

rrrr

rrrrrr

rrrrr

F

O

ri

rj

ri-rj

qi

qj

Fij

Principle of Superposition• Total force on one charge i is

• i.e. linear superposition of forces due to all other charges

• Test charge: one which does not influence other ‘real charges’ – samples the electric field, potential

• Electric field experienced by a test charge qi ar ri is

ij

ij2ij

j

oii r

q4

1q rF ˆ

ij

ij2ij

j

oi

iii r

q4

1q

rFrE ˆ

Electric Field• Field lines give local direction of field

• Field around positive charge directed away from charge

• Field around negative charge directed towards charge

• Principle of superposition used for field due to a dipole (+ve –ve charge combination). Which is which?

qj +ve

qj -ve

Flux of a Vector Field• Normal component of vector field transports fluid across

element of surface area

• Define surface area element as dS = da1 x da2

• Magnitude of normal component of vector field V is V.dS = |V||dS| cos()

• For current density jflux through surface S is

Cm2s-1

da1

da2

dS

dS = da1 x da2

|dS| = |da1| |da2|sin(/2)

dS`

S surface closed

.dSj

• Electric field is vector field (c.f. fluid velocity x density)• Element of flux of electric field over closed surface E.dS

da1

da2

n

Flux of Electric Field

ˆˆˆ

ˆˆ

ˆ

θn

naaSa

θa

x

d dθ sinθ r d x dd

d sinθ rd dθ rd

221

2

1

o

oo

22

o

q.d

d4

qd dθ sinθ4

q

1 d dθ sinθr .r4

q.d

S

SE

n.r nrSE ˆˆˆˆ

Gauss’ Law Integral Form

• Factors of r2 (area element) and 1/r2 (inverse square law) cancel in element of flux E.dS

• E.dS depends only on solid angle d

da1

da2

n

Integral form of Gauss’ Law

o

ii

o

21

q.d

d4

qq.d

S

SE

SE

Point charges: qi enclosed by S

q1

q2

v withincharge total)d(

)dv(.d

V

o

V

vr

rSE

S

Charge distribution (r) enclosed by S

Differential form of Gauss’ Law• Integral form

• Divergence theorem applied to field V, volume v bounded by surface S

• Divergence theorem applied to electric field E

V

SS

dv .ddS. VSV. nV

V.n dS .V dv

o

V

)d(.d

rrSE

S

VV

V

)dv(1dv .

dv .d

rE

ESE.

o

So

)( )(. rrE

Differential form of Gauss’ Law(Poisson’s Equation)

Apply Gauss’ Law to charge sheet• (C m-3) is the 3D charge density, many applications make

use of the 2D density (C m-2):

• Uniform sheet of charge density Q/A• By symmetry, E is perp. to sheet• Same everywhere, outwards on both sides• Surface: cylinder sides + faces • perp. to sheet, end faces of area dA • Only end faces contribute to integral

+ + + + + ++ + + + + +

+ + + + + ++ + + + + +

E

EdA

ooo

2 ESE.

S

.dAE.2dAQd encl

• ’ = Q/2A surface charge density Cm-2 (c.f. Q/A for sheet)• E 2dA = ’ dA/o

• E = ’/2o (outside left surface shown)

Apply Gauss’ Law to charged plate

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

E

dA

• E = 0 (inside metal plate)• why??

+

+

+

+

+

+

+

+• Outside E = ’/2o + ’/2o = ’/o = /2o

• Inside fields from opposite faces cancel

Work of moving charge in E field• FCoulomb=qE• Work done on test charge dW • dW = Fapplied.dl = -FCoulomb.dl = -qE.dl = -qEdl cos • dl cos = dr

• W is independent of the path (E is conservative field)

A

B

q1

q

rr1

r2E

dl

B

A

21o

1

r

r 2o

1

2o

1

.dq

r1

r1

4qq

drr1

4qqW

drr1

4qqdW

2

1

lE

0path closed any

lE.d

Potential energy function• Path independence of W leads to potential and potential

energy functions

• Introduce electrostatic potential

• Work done on going from A to B = electrostatic potential energy difference

• Zero of potential energy is arbitrary– choose (r→∞) as zero of energy

r1

4q)(

o

1

r

B

A

BA

.dq

)(-)(q)PE(-)PE(W

lE

ABAB

Electrostatic potential• Work done on test charge moving from A to B when charge

q1 is at the origin

• Change in potential due to charge q1 a distance of rB from B

Bo

1

r

2o

1

B

r1

4q)(

drr1

4q

.d

-)()(-)(

B

B

lE

BAB

0

)(-)(q)PE(-)PE(WBA ABAB

Electric field from electrostatic potential

• Electric field created by q1 at r = rB

• Electric potential created by q1 at rB

• Gradient of electric potential

• Electric field is therefore E= –

3o

1

r4q rE

r1

4qr

o

1B

)(

3o

1B r4

qr r

)(

Electrostatic energy of point charges• Work to bring charge q2 to r2 from ∞ when q1 is at r1 W2 = q2 2

• NB q2 2 = q1 1 (Could equally well bring charge q1 from ∞)• Work to bring charge q3 to r3 from ∞ when q1 is at r1 and q2 is at r2

W3 = q3 3

• Total potential energy of 3 charges = W2 + W3

• In general

O

q1 q2

r1 r2

r12

12o

12 r

1q

4

O

q1 q2

r1 r2

r12

r3

r13

r23

23o

2

13o

13 r

1qr1q

44

ji j ij

ji

ji j ij

ji r

qq1

21

rq

q1Woo 44

Electrostatic energy of charge distribution

• For a continuous distribution

space allspace allo

space allo

space all

)(d)(d4

121W

)(d4

1)(

)()(d21W

r'rr'r'r r

r'rr' r'r

rr r

Faraday's Law• 

• Magnetic field around a permanent magnet.

B

•Magnetic field around a straight conductor carrying a steady current I.

•Magnitude of B is directly proportional to the current I value and inversely proportional to the distance from the conductor.

Magnetic flux

S

B SdB

cos S

B dsB

211 mTWb

WbB

Faraday’s Law

md d dxBl x Bldt dt dt

dxBlv Bldt

E

mTherefore, ddt

E

• CONCLUSION: to produce emf one should make ANY change in a magnetic flux with time!

•Consider the loop shown:

LENZ’S Law •The direction of the The direction of the emf induced by emf induced by changing flux will changing flux will produce a current produce a current that generates a that generates a magnetic field magnetic field opposing the flux opposing the flux change that change that produced it.produced it.

Lenz’s Law

•B, H

•Lenz’s Law: emf appears and current flows that creates a magnetic field that opposes the change – in this case an increase – hence the negative sign in Faraday’s Law.

•B, H

•N •S

•V-, V+

•Iinduced

Faraday’s Law for a Single Loop

dtdE

Faraday’s Law for a coil having N Faraday’s Law for a coil having N turnsturns

dtdNE

Lenz's Law

Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated.

• Why???– If current reinforced the change, then

the change would get bigger and that would in turn induce a larger current which would increase the change, etc..

– No perpetual motion machine!

Conclusion: Lenz’s law results from energy conservation principle.

•In 1831 Joseph Henry discovered magnetic induction.

The History of Induction

•Joseph Henry•(1797-1878)

•Michael Faraday•(1791-1867)

• Michael Faraday's ideas about conservation of energy led him to believe that since an electric current could cause a magnetic field, a magnetic field should be able to produce an electric current. He demonstrated this principle of induction in 1831.

•So the whole thing started 176 years ago!

THANK YOU