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lecture 6 - Chapter 26 Electricity and Magnetism Electric...

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1 Electricity and Magnetism Electric Field: Continuous Charge Distribution Chapter 26
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2 2 2 0 ˆ ˆ lim i i e i e q i i q dq k k r r = = E r r Find electric field at point P. + + + + + + + + i i E E = r r P 2 ˆ i i e i i q E k r r = r r Continuous Charge Distribution
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3 Electric Field: Continuous Charge Distribution + 1 E r 2 E r Electric field 1 2 E E E = + r r r 2 ˆ e q E k r r = r r + + 1 E r 2 E r + + 3 E r + 1 E r 2 E r + + + + + + + + + + + + + + i i E E = r r In this situation, the system of charges can be modeled as continuous The system of closely spaced charges is equivalent to a total charge that is continuously distributed along some line, over some surface, or throughout some volume
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4 Electric Field: Continuous Charge Distribution The total electric charge is Q. What is the electric field at point P ? Q P Q P Q P linear surface volume
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5 Continuous Charge Distribution: Charge Density The total electric charge is Q. Q Q Q Linear, length L Surface, area A Volume V Amount of charge in a small volume dl: Q dq dl dl L λ = = Q L λ = Linear charge density Amount of charge in a small volume dA: Q dq dA dA A σ = = Q A σ = Surface charge density Amount of charge in a small volume dV: Q dq dV dV V ρ = = Q V ρ = Volume charge density
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6 Electric Field: Continuous Charge Distribution Procedure : Divide the charge distribution into small elements, each of which contains Δq Calculate the electric field due to one of these elements at point P Evaluate the total field by summing the contributions of all the charge elements Symmetry : take advantage of any symmetry to simplify calculations 2 ˆ e q k r = E r For the individual charge elements 2 2 0 ˆ ˆ lim i i e i e q i i q dq k k r r = = E r r Because the charge distribution is continuous
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7 Electric Field: Symmetry + + P E r E r P P x no symmetry no symmetry no symmetry
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8 Electric Field: Symmetry + 1 2 1 E r + 2 E r Electric field 1 10 qμC = 2 10 qμC = 5 m 6 m 5 m 1 2 E E E = + r r r 2 ˆ e q E k r r = r r E r 2 E r ϕ ϕ 1 2 cos E = 1, 2, 2 2 z z E E E = = 1, z E r + + + + 1 E r 1, z E r E r 1, 2 4 4 cos z E E = = 1, 1 cos z E = ϕ ϕ The symmetry gives us the direction of resultant electric field
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9 Electric Field: Continuous Charge Distribution O R o 90 P λ - linear charge density h What is the electric field at point P ?
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