The figure below shows a battery connected across a uniform resistor R 0 . A sli

Question

The figure below shows a battery connected across a uniform resistor R0. A sliding contact can move across the resistor from x = 0 at the left to x = x0 at the right. Moving the contact changes how much resistance is to the left of the contact and how much is to the right.

Find the rate at which energy is dissipated in resistor R as a function of x. Denote the emf as ?.

The figure below shows a battery connected across a uniform resistor R0. A sliding contact can move across the resistor from x = 0 at the left to x = x0 at the right. Moving the contact changes how much resistance is to the left of the contact and how much is to the right. Find the rate at which energy is dissipated in resistor R as a function of x. Denote the emf as ?.

Explanation / Answer

from the given problem we can see that the part of Ro connected in paralle with R is given by

The resistance R1 can be expressed as,

             R1 = Ro x / L,   here, L = xo

the volatge across the ressitor is,

               VR = eR'/Req,    here, e is the emf.

Then, R ' = R R1 / (R + R1)

So, the equivalent resistance is,

   Req = Ro [1 - (x / L)] +R'

Hence, the power can be expressed as,

   PR = VR2 /R

         = (1 / R){[(eR R1) / (R + R1)] / [Ro(1 - (x / L) + [R R1 / R + R1]}2

         = (1 / R){[(eR R1) / (R + R1)] / [Ro(1 - (x / x0) + [R R1 / R + R1]}2

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