1. TOPIC : GAUSS;S LAW AND APPLICATION
Mahatma Phule Arts, Science And Commerce College, Panvel
( Academic Year 2015 – 2016 )
SEMESTER 5TH
T. Y. B. Sc.
Physics Paper FOURTH
ELECTRODYNAMICS
TOPIC : GAUSS;S LAW AND APPLICATION By
Dr. Sarode M. T.
Department of Physics
05/08/2015

2. 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 O ri rj
ri-rj qi qj
Fij
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

3. 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
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

4. Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel
Electric Field
qj +ve
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
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

5. Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel
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(Y)
For current density j
flux through surface S is
Cm2s-1
da1
da2 dS
dS = da1 x da2
|dS| = |da1| |da2|sin(p/2) Y
dS`
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

6. Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel
Flux of Electric Field
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 q f
Gauss’ Law Integral Form
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

7. Integral form of Gauss’ Law
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 dW
da1
da2 n q f
Point charges: qi enclosed by S q1 q2
Charge distribution r(r) enclosed by S
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

8. 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.n dS
.V dv
Differential form of Gauss’ Law
(Poisson’s Equation)
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

9. Apply Gauss’ Law to charge sheet
r (C m-3) is the 3D charge density, many applications make use of the 2D density s (C m-2):
Uniform sheet of charge density s = 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 dA
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

10. Apply Gauss’ Law to charged plate
s’ = Q/2A surface charge density Cm-2 (c.f. Q/A for sheet)
E 2dA = ’ dA/o
E = ’/2o (outside left surface shown) E dA
E = 0 (inside metal plate)
why??
Outside E = ’/2o + ’/2o = ’/o = /2o
Inside fields from opposite faces cancel
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

11. Work of moving charge in E field
04/09/2013
Work of moving charge in E field
FCoulomb=qE
Work done on test charge dW
dW = Fapplied.dl = -FCoulomb.dl = -qE.dl = -qEdl cos q
dl cos q = dr
W is independent of the path (E is conservative field) A B q1 q r r1 r2 E dl
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

12. 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 f(r→∞) as zero of energy
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

13. 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
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

14. 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= – f
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

15. Electrostatic energy of charges
In vacuum
Potential energy of a pair of point charges
Potential energy of a group of point charges
Potential energy of a charge distribution
In a dielectric (later)
Potential energy of free charges
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

16. Electrostatic energy of point charges
04/09/2013
Electrostatic energy of point charges
Work to bring charge q2 to r2 from ∞ when q1 is at r1 W2 = q2 f2
NB q2 f2 = q1 f1 (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 f3
Total potential energy of 3 charges = W2 + W3
In general O q1 q2 r1 r2
r12 O q1 q2 r1 r2
r12 r3
r13
r23
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

17. Electrostatic energy of charge distribution
For a continuous distribution
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

18. Energy in vacuum in terms of E
Gauss’ law relates r to electric field and potential
Replace r in energy expression using Gauss’ law
Expand integrand using identity:
.F = .F + F.
Exercise: write  = f and F = f to show:
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

19. Energy in vacuum in terms of E
For pair of point charges, contribution of surface term
  1/r   -1/r2 dA  r2 overall  -1/r
Let r → ∞ and only the volume term is non-zero
Energy density
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel

20. Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel
THANK YOU
Dr.M.T.Sarode, dept. of Physics, M.P.A.S.C.College,Panvel