CHAPTE () Electric Chrges, Electric Chrge Densities nd Electric Field Intensity
Chrge Configurtion ) Point Chrge: The concept of the point chrge is used when the dimensions of n electric chrge distriution re very smll compred to the distnce to the neighoring chrges, i.e. the point chrge is occupying very smll physicl spce ) Distriuted Chrge The chrge my e distriuted long line, mong surfce or mong volume. () Line Chrge: The line chrge density l is defined s the chrge per unit length. l lim l Q l C m dl + + + + + + + + + + + + + + + + dq L dl So, l dq dl C m dq l dl The totl chrge Q of the line cn e determined y
Q dq l dl l l [C] For uniform line chrge, where l is constnt Q l l [C] U() Surfce Chrge: The surfce chrge density s is defined s the chrge per unit surfce re. s lim s Q s C m s dq ds C m So, dq s ds The totl chrge Q of the surfce cn e determined y
Q dq s ds For uniform surfce chrge, where s is constnt Q s S [C] U) Volume chrge density ( v ) v lim v Q v C m v dq dv C m dv dq v dv So, dq v dv The totl chrge Q of the surfce cn e determined y Q dq v dv UFor uniform volume chrgeu, where v is constnt Q v V [C]
Exmple: A uniform sphericl volume chrge density distriution contins totl chrge of -8 C, if the rdius of the sphere x - m. Find v. Solution: Q 8 v C Q V, r *. 8 4π *8 * 8 m, V.98 * 4π r 4 Cm 4π ( * ) 8 * 6 m UExmple: A non-uniform sphericl volume chrge density distriution with v k r s C.mP - P. Find the totl chrge contined in the volume of the sphere of rdius [m] USolution: dq v dv Q dq v dv k r s r s sinθ dθ dφ
to Q k r s [φ] π [ cos θ ][ cos θ ] π Q πk C Coloum s Lw: Force Between Two Point Chrges: The force etween two sttionry point chrges Q,Q is proportionl to the product of the two chrges nd inversely proportionl to the squre of the distnce etween them. F â Q Q F F Q Q 4πε o F Q Q 4πε o Where: unit vector from chrge Q chrge. Q
P is P nd, ε o 9 6π 8. 85x F/m Force on Point Chrge due to n Point Chrges: Force F t on point chrge Q t due to n point chrges cn e determined s n F t i Q i Q t 4πε o it it [N] Q t t n t Qt UExmple: Find the force F in vcuum on point Q P C due to -5 point chrge Q*P C where centered t point Q (,,)m nd t (,4,5) Q USolution: F Q Q 4πε QQ 4πε ( ) xˆ + ( 4) yˆ + ( 5) zˆ xˆ yˆ ˆ z Q n -4 4 + 9 + 9
ˆ xˆ yˆ ˆ z 4 5 ( ) ( xˆ yˆ ˆ) z F N 4π 8.85 ( ) Electric Field Intensity t Point due to Point Chrge Q: It is vector force cting on unit (+ve) chrge. The electric field intensity due to point locted t distnce from the chrge Q is given y: E Q 4πε o Q 4πε o volt meter (V/m) Electric Field Intensity t Point due to Point Chrges Q, Q,.., Q n : If we hve system of chrges Q, Q Q n. the totl electric field t point is the vector sum of ll fields due to the different chrges. n Q i E t i [N/C] 4πε o it α n Q i E t 4πε o i i i [N/C] Q â â Q n â n Q n
Exmple: Find the electric field intensity t the point (,4,5) m, due to point chrge Q * -5 C, locted t (,,) m. Solution: E Q 4πε ( ) xˆ + (4 ) yˆ + (5 ) zˆ xˆ + yˆ + z 4 + 9 + 9 E 5 4π (8.85 (xˆ + yˆ + ˆ) z )( )... xˆ +... yˆ +... zˆ
Electric Field Intensity t point p (r c, φ, z) due to line chrge de dq 4πε o z de dq 4πε o x dl y E l dl 4πε o Electric Field Intensity t Point p (r c, φ, z) due to Uniform Line Chrge Along z-axis z dz' z' z α α (r c,φ,z) y z r c φ x
de dq 4πε o dq l dl l dz rc r c z (z z) r c + (z z) r c r c z (z z) r c + (z z) de l dz rc r c z (z z) 4πε o r c + (z z) note: E l rc r c z z z 4πε o r c + z z dz dx c + x x c c + x
xdx c + x c + x E l 4πε o rc r c z dz r c + z z z z dz r c + z z E l 4πε o rc r c + z (z z) r c [r c + (z z) ] [r c + (z z) ] E l r c ( z) 4πε o r c [r c + ( z) ] r c + z r c [r c + ( z) ] ( z) [r c + ( z) ] r c [r c + ( z) ]
E l 4πε o r c r c [sin α + sin α ] + z r c [cos α cosα ] E l 4πε o r c rc [sin α + sin α ] + z [cos α cosα ] Note: - For infinite line α α 9 o So, E rc l πε o r c Projection point rc r c Intersection point
Electric Field Intensity t point p (,,z) on the xis of ring chrged with uniform L of rdius centered t the origin nd positioned in (x-y) plne d E Ẑ (,, z) dφ Yˆ dl dφ dq dl l dφ l Xˆ de dq 4πε o dq l dl l r c dφ l dφ rc r c + z z rc + z z r c + z + z
r c + z z + z de l dφ rc + z z 4πε o [ + z ] E l 4πε o π rc + z z + z dφ E π l 4πε o [ + z ] r dφ c π + z z dφ Since, the unit vector rc is not constnt unit vector nd it is function of φ, nd since So, E l 4πε o [ + z ] rc x cosφ + y sinφ π x cosφ dφ π + z z dφ π y sinφ dφ
E π l zz dφ 4πε o [ + z ] E z l z ε o [ + z ] Exmple: A uniform line chrge of infinite extent with l nc/m lies on z-xis. Find E t (6,8,) m. Solution: E l πε r c r c x + y E rˆ c 6 + 8 9 rˆ c 6ˆ rc V / m π (8.85* )
Electric field intensity of surfce chrge: d E z (,,z ) S r c ds y x de dq 4πε o dq s ds s r c dr c dφ rc r c + z z r c + z
r c r c + z z r c + z de s r c dr c dφ rc r c + z z 4πε o r c + z E s r c dr c 4πε o r c +z π rc dφ + r c dr c r c +z π z z dφ Since, the unit vector rc is not constnt unit vector nd it is function of φ, nd since So, E s 4πε o + rc x cosφ + y sinφ r c dr c r c + z [r c ]dr c π x cosφ y sinφ dφ π r c + z z z dφ Then,
E s 4πε o nd since, r c dr c r c +z [r c ]dr c r c + z r c +z π z z dφ + [ +z ] [z ] So, E z s ε o z [ + z ] + z z E z s ε o z + z For infinite surfce E z s ε o Note: For infinite surfce the electric field E z s ε o nd in direction norml to the surfce nd out of it.
Exmple: Two infinite uniform sheets of chrge s nd s locted t x ± s shown in figure. Find the electric field in ll regions. Solution: [ ]x s s E E E x s E x s E t ˆ ˆ, ˆ : egion ε ε ε + + [ ]x s s E x s E x s E t ˆ ( ˆ), ˆ) ( egion : ε ε ε + x x S y - s
egion : E E t s xˆ, E ε [ + ]xˆ s s ε s ε xˆ