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Electric Field of a Uniform Ring of Charge
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[QUOTE="kuruman, post: 6826740, member: 192687"] My personal preference is not to use constants of integration in physics problems and use instead upper and lower limits of integration. Consider, for example, the simple question of finding the velocity as a function of time for an object that moves under constant acceleration. One has to integrate both sides of the equation ##dv=a~dt ## where ##a## is constant. One might write the mathematically correct expression that is taught in intro calculus in terms of a constant of integration,$$v=at+C.$$ Because the equation involves physical entities associated with the physical motion of an object, I view the integration as adding up infinitesimal elements of velocity on the LHS and acceleration times elements of time on the RHS. As with any addition, you have to start and end somewhere and that's where the limits of integration come in. I start the addition when the clock timing the motion reads ##t_0##, at which time the velocity is ##v_0##, and stop adding when the clock reds ##t## and the velocity is ##v##. This is addition is written symbolically as $$ \int_{v_0}^{v}dv=a\int_{t_0}^{t}dt$$ from which $$v-v_0 = a(t-t_0). $$ With ##t_0=0##, which is usually the case, and after some rearrangement one gets the familiar equation $$v=v_0+at.$$ The integration constant aficionados will correctly tell you that this is the same as setting the integration constant ##C=v_0##. I prefer using definite integrals because it reminds me that integration is nothing but addition and because any needed integration constant is taken into account automatically. [/QUOTE]
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Electric Field of a Uniform Ring of Charge
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