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Resistance paradox

  1. Mar 1, 2004 #1
    Hi folks,

    Please, please help me, i know the answer to my question must have an easy answer but i can't figure it out.
    Consider a simple circuit with a piece of copper wire, a switch and a battery. I know that when i close the switch the PD drives the electrons around the wire at a rate which is dependant on the total resistance of the wire. Heat will be produced as the electrons accelerate for short periods of time, collide with the copper atoms, then re-accelerate etc etc.
    Now this makes me think that if the resistance of the wire was greater then there would be more collisions and hence more heat produced but when i do the calculations eg P=VI i can deduce that the opposite is true i.e. more heat is produced if i lower the resistance because the current will increase and thus the power will increase. This also makes sense to me as i can imagine as the electrons speed up they will collide with atoms at greater force and hence 'produce' more heat. The thing is i know that the latter is correct by experiment but can someone explain to me in easy terms what is happening and where i am going wrong in my thoughts?

    Thanks in advance
     
  2. jcsd
  3. Mar 1, 2004 #2

    Cliff_J

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    Well, your current and resistance are in a linear relationship but the power is exponential.

    So with a fixed voltage from a battery, examining current^2 * resistance = power you can quickly see the significance on the heat produced by changing the current compared to changing the resistance.

    Cliff
     
  4. Mar 1, 2004 #3
    Cheers, that makes perfect sense, so in a way would you say my assumption was correct i.e. raising the resistance does increase the power output but doing this in reality means current is reduced and hence the apparant paradox is you can't raise the resistance AND maintain the current, all other things equal??

    By the way is my idea of how the electrons behave under the influence of the voltage correct??

    Thanks again

    Beckett
     
  5. Mar 1, 2004 #4

    Integral

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    I think much of your problem is coming from the mental model you have of current flow. Electrons do not ever move very fast or very far. Compared to the signal velocity, electron drift velocity is very low in normal circuit conditions. What is more important then the electron velocity is the NUMBER of collisions occurring. The higher the current flow the MORE collisions there will be thus more power is required and more power is dissipated.
    Even this collision model may fail in some circumstances because we are applying macroscopic concepts to the subatomic world, the only way to fully understand the mechanisms occurring at this level is through the eyes of Quantum Mechanics. Just keep in mind that electrons are NOT hard little billiard balls bouncing along through the wire.
     
  6. Mar 1, 2004 #5

    Cliff_J

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    Ok, I think you overlooked ohm's law and that's where you arrived at the paradox.

    Remember, voltage = current * resistance. We have a fixed voltage with a battery. But we can determine that if current increases by a percentage, resistance will decrease by the same percentage. If I double the resistance, I end up with half the current.

    But power is different. If I double the resistance, I get 1/4 the power. If I triple the resistance, I get 1/9 the power.

    Maximum power occurs near the zero resistance point as current goes to infinity, minimum power occurs when the current goes to zero and the resistance goes to infinity.

    Cliff
     
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