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CourseNotes.4

# CourseNotes.4 - 3.3 The Electric Field for Continuous...

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3.3 The Electric Field for Continuous Charge Distributions For a continuous charge distribution, we can write down the differential bit of electric field due to a differential bit of charge as: dE = 1 4 π 0 dq r 2 (6) Using this knowledge, we can find the total electric field by summing up (integrating) all of the contributions from the differential elements of charge. The challenge in doing so usually comes down 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 express dq in terms of an integration variable. For straight line charges: dq = λ dx and for circular line charges: dq = λ R dθ Using the second form, we can derive the electric field due to a ring of charge as measured on the axis of the ring: E = 1 4 π 0 qz ( z 2 + R 2 ) 3 / 2 (7) Similarly, for a charged surface such as a disk or a plane, we can use the surface charge density σ to express dq in terms of an integration variable. In the case of a charged disk: dq = σ dA = σ (2 πr ) dr
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