Gauss's Law: Applications and Examples in Electromagnetism, Schemes and Mind Maps of Electrical and Electronics Engineering

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¨ to find electric field intensity

¨ simpler than Coulomb’s law

¨ symmetrical charges distribution

¨ obtained from electric flux density

D Gauss’s Law E D

+Q +Q

• All charge transfer to the

imaginary surface through

the flux lines

• Total Y E

from the inner

sphere equal its total charge,

Q

• Magnitude of charge on the

imaginary surface =

magnitude of charge inside

the surface = Q

 
  Imaginary surface

Ε 2 1 2 3 Ε = E ¹ E

Gaussian surfaces

+Q

1 Ε

D

2

E

2

E

3 2 D E 2 0 = e

Gauss’s Law

The integration of electric flux density, D over

a closed Gaussian surfaced S equal to the

charge enclosed the surface.

enc S

D • ds = Q

ò

Example 1: Find the total flux 1 meter from a 12 nC point charge located at the origin.

Total flux = total charge inside

the imaginary closed surface

= 12nC

Example 3: From the same sphere as in example 2, find the the total flux at (a) r = 5 meters from the center of the sphere.

(a) Total flux = total charge enclosed = 0

r

s

Example 3: From the same sphere as in example 2, find the the total flux at (b) r = 5 meters from the surface of the sphere. 5

r

s

Example 5: Using Gauss, find the electric field intensity around an infinite surface of charge distribution rS. Example 6: A line charge of rL= 20 nC/m is parallel to x-axis, located at y = - 3, z = - 4. Find the electric field intensity at the origin. Example 7: The volume in a cylinder between r = 2 and r = 4 meters contains a uniform charge density of rV = 10 nC/m 3

. Find the electric flux density and field intensity in all regions.

Example 5: Using Gauss, find the electric field intensity around an infinite surface.

Infinite surface

s

1

s

2

S

r

S 1 2 1 2 ˆ ˆ ˆ ˆ 1 2 1 2 , 2 2 z z S S S S z z S z S S z zD zds zD zds ds D S D S S S S S D S S D r r r r

• • • • - =
    • = = = = = ò ò ò Infinite surface: S1 = S2 = S

2 ˆ (^) , 0 2 ˆ (^) , 0 2 S z S S D z z D z z r r r = ì

ï ï = (^) í ï

• < ïî ˆ 2 S infinite surface D n r =

S

r At z = 0 S n^ ˆ E n V m S inf inite surface / 2 ˆ 0 e r =

Divergence = outward flux from a closed surface

as the volume shrinks to zero.

D D (^) y D x z div D D v x y z r æ ö ç ÷ = ç^ ÷ ç ÷ ç ÷ è ø ¶ ¶ (^) ¶ =Ñ× = + + ¶ ¶ ¶ v v S v (^) v Q v D d s div D = r D = D

º D ® D ® ò 0 0 lim lim Divergence

(a) (b) (^) (c)

(a) positiive divergence (b) negative divergence

(c) zero divergence

Divergence