3.3The Electric Field for Continuous Charge DistributionsFor a continuous charge distribution, we can write down the differential bit of electric field due toa differential bit of charge as:dE=14π0dqr2(6)Using this knowledge, we can find the total electric field by summing up (integrating) all of thecontributions from the differential elements of charge.The challenge in doing so usually comesdown to expressing the charge in a way such that it can be integrated over.For line charges we can use the linear charge density to expressdqin terms of an integrationvariable. For straight line charges:dq=λ dxand for circular line charges:dq=λ R dθUsing the second form, we can derive the electric field due to a ring of charge as measured on theaxis of the ring:E=14π0qz(z2+R2)3/2(7)Similarly, for a charged surface such as a disk or a plane, we can use the surface charge densityσto expressdqin terms of an integration variable. In the case of a charged disk:dq=σ dA=σ(2πr)dr
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