notes3 - 9/21/01 Examples of Gauss' Law Example Determine...

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1 9/21/01 Examples of Gauss' Law Example Determine the net electric flux out of the each of the surfaces shown in the figure. Solution: DS S dQ 1 zz ⋅= S Q 2 0 zz = S Q Q Q 3 zz + =
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2 It is interesting to note that the integral form of Gauss' law can be used to solve for the E or D in problems where there is a high degree of symmetry (typically 2 degrees of symmetry). The next three examples demonstrate the application of Gauss' law to find the E associated with three different charge distributions. Example Find the electric field due to a uniformly distributed sphere of charge ( ρ V is a constant for r a ). Solution: Using Gauss' law V V S dd V ⋅= ∫∫ ∫∫∫ DS w By symmetry ˆ r D = Dr , where D r is only a function of r . Thus, assuming r a
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3 22 00 000 2 3 2 2 2 2 2 ˆ sin sin 4 sin 3 sin 4 4 a V rr rV r r r rd d r d r d d a Dr d d Q DrQ Q D r ππ θ θφ ρ π θθφ ρ θθφ = ⋅= = = = = ∫∫ ∫∫∫ or E a r r V = ε 3 2 3 . Note that this is the same D field as that for a point charge Q at the origin. For r a , 3 2 ˆ sin sin 4 4 3 3 r V V r V r d r d r d d r r D ρπ = = = or E Qr a r = 4 3 πε .
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4 Example Find the electric field due to an infinite uniform line charge. Solution: Using Gauss' law S dd ρ ⋅= ∫∫ DS A A A w By symmetry ˆ D = D ρ , where D is only a function of . Thus, 12 3 ˆ ˆˆ () ( ) zh SS S ds ds ds d ρρ == = + ⋅+ + = ∫∫ ∫∫ ∫∫ D ρ Dz D z A A A 2 0 (2 )2 2 2 hh Dd z d d z Dh h E π ρφ πρ περ −− = = = ∫∫ A A A This solution agrees with our previous result.
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5 Static Charges and Conductors Within and on conductors, charges can move with little resistance to motion. If free charges are placed within a conductor there will be an electric field established between them.
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notes3 - 9/21/01 Examples of Gauss' Law Example Determine...

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