1. EEE PPT
• NAME:HARSHIL.R.SHAH
• EN NO:151080106025
• BRANCH:CIVIL
• SEM:2
• TOPIC;OHM’S LAW,GAUSS LAW,
FARADE’S LAW
• SUMBITED BY
2. 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
Opposition
Cause
Effect =
3. 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
4. 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.
5. Ohm’s Law
Where: I = current (amperes, A)
E = voltage (volts, V)
R = resistance (ohms, Ω)
R
E
I =
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
ji
ijjiijjiij
ij2
ij
ji
o
ji3
ji
ji
o
ij
r
r
qq
4
1qq
4
1
rr
rr
rrrrrr
rrr
rr
F
−
−
=−=−=
=−
−
=
πεπε
O
ri
rj
ri-rj
qi
qj
Fij
9. 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
ij2
ij
j
o
ii
r
q
4
1
q rF ˆ
πε
( ) ∑≠
==
ij
ij2
ij
j
oi
i
ii
r
q
4
1
q
r
F
rE ˆ
πε
10. 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
11. 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(Ψ)
• For current density j
flux through surface S is
Cm2
s-1
da1
da2
dS
dS = da1 x da2
|dS| = |da1| |da2|sin(π/2)
Ψ
dS`
∫ Ssurfaceclosed
.dSj
12. • 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
θ
φ
Flux of Electric Field
ϕ
ϕ
ϕϕ
ˆˆˆ
ˆ
ˆ
ˆ
θn
naaS
a
θa
x
ddθsinθrdxdd
dsinθrd
dθrd
2
21
2
1
=
==
=
=
o
oo
2
2
o
q
.d
d
4
q
ddθsinθ
4
q
1ddθsinθr.
r4
q
.d
ε
πε
ϕ
πε
ϕ
πε
∫ =
Ω==
==
S
SE
n.rn
r
SE ˆˆˆ
ˆ
Gauss’ Law Integral Form
13. • 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 dΩ
da1
da2
n
θ
φ
Integral form of Gauss’ Law
o
i
i
o
21
q
.d
d
4
qq
.d
ε
πε
∑
∫ =
Ω
+
=
S
SE
SE
Point charges: qi enclosed by S
q1
q2
vwithinchargetotal)d(
)dv(
.d
V
o
V
=
=
∫
∫
∫
vr
r
SE
ρ
ε
ρ
S
Charge distribution ρ(r) enclosed by S
14. 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
ε
ρ∫
∫ =
rr
SE
S
∫∫
∫∫
=∇
∇=
VV
V
)dv(
1
dv.
dv.d
rE
ESE.
ρ
εo
S
oε
ρ )(
)(.
r
rE =∇
Differential form of Gauss’ Law
(Poisson’s Equation)
15. Apply Gauss’ Law to charge sheet
• ρ (C m-3
) is the 3D charge density, many applications make use
of the 2D density σ (C m-2
):
• Uniform sheet of charge density σ = 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 ε
σ
ε
σ
ε 2
=⇒=⇒=∫ ESE.
S
.dA
E.2dA
Q
d encl
16. • σ’ = Q/2A surface charge density Cm-2
(c.f. Q/A for sheet)
• E 2dA = σ’ dA/εo
• E = σ’/2εo (outside left surface shown)
Apply Gauss’ Law to charged plate
++++++
++++++
++++++
++++++
E
dA
• E = 0 (inside metal plate)
• why??
++++
++++
• Outside E = σ’/2εo + σ’/2εo = σ’/εo = σ/2εo
• Inside fields from opposite faces cancel
17. Work of moving charge in E field
• FCoulomb=qE
• Work done on test charge dW
• dW = Fapplied.dl = -FCoulomb.dl = -qE.dl = -qEdl cos θ
• dl cos θ = dr
A
B
q1
q
r
r1
r2
E
dl
θ
∫
∫
−=
−−=
−=
−=
B
A
21o
1
r
r 2
o
1
2
o
1
.dq
r
1
r
1
4
q
q
dr
r
1
4
q
qW
dr
r
1
4
q
qdW
2
1
lE
πε
πε
πε
0=∫ pathclosedany
lE.d
18. 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 φ(r→∞) as zero of energy
r
1
4
q
)(
o
1
πε
φ =r
( )
∫−=
==
B
A
BA
.dq
)(-)(q)PE(-)PE(W
lE
ABAB φφ
19. 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
2
o
1
B
r
1
4
q
)(
dr
r
1
4
q
.d
-)()(-)(
B
πε
φ
πε
φφφ
=
−=
−=
=∞→
∫
∫
∞
∞
B
lE
BAB 0
( ))(-)(q)PE(-)PE(WBA ABAB φφ==
20. 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= – φ
3
o
1
r4
q r
E
πε
=
r
1
4
q
r
o
1
B
πε
φ =)(
3
o
1
B
r4
q
r
r
πε
φ −=∇ )(
21. Electrostatic energy of point charges
• Work to bring charge q2 to r2 from ∞ when q1 is at r1 W2 = q2 φ2
• NB q2 φ2 =q1 φ1 (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 φ3
• Total potential energy of 3 charges = W2 + W3
• In general
O
q1
q2
r1 r2
r12
12o
1
2
r
1q
πε
ϕ
4
=
O
q1
q2
r1 r2
r12
r3
r13
r23
23o
2
13o
1
3
r
1q
r
1q
πεπε
ϕ
44
+=
∑ ∑∑ ∑ ≠<
==
ji j ij
j
i
ji j ij
j
i
r
q
q
1
2
1
r
q
q
1
W
oo πεπε 44
22. Electrostatic energy of charge
distribution
• For a continuous distribution
∫∫
∫
∫
−
=
−
=
=
spaceallspaceallo
spaceallo
spaceall
)(
d)(d
4
1
2
1
W
)(
d
4
1
)(
)()(d
2
1
W
r'r
r'
r'rr
r'r
r'
r'r
rrr
ρ
ρ
πε
ρ
πε
φ
φρ
27. Faraday’s Law
md d dx
Blx Bl
dt dt dt
Φ
= =
dx
Blv Bl
dt
= =E
m
Therefore,
d
dt
Φ
=E
• CONCLUSION: to produce emf one should make
ANY change in a magnetic flux with time!
•Consider the loop
shown:
28. LENZ’S Law
•The direction of theThe direction of the
emf induced byemf induced by
changing flux willchanging flux will
produce a currentproduce a current
that generates athat generates a
magnetic fieldmagnetic field
opposing the fluxopposing the flux
change thatchange that
produced it.produced it.
29. 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
31. Faraday’s Law for a coil having NFaraday’s Law for a coil having N
turnsturns
dt
d
NE
Φ
−=ε=
32. 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.
33. •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!