I learnt from my Physics class that voltage = integral of E*dl, which clearly indicates that voltage is distance dependent. But why do we say that a battery has a fixed voltage? Won't a battery's voltage vary depending on the length of the wire connecting the cathode to the anode?
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Kinetic energy is given by $\int F~dl$, and so is – Chris Jan 22 '18 at 05:14
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A battery doesn't work by (or have the effect of) setting up a constant electric field between its terminals. – The Photon Jan 22 '18 at 05:29
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The equation applies to an electrical field between two areas/plates with different electric potentials, like the individual cells inside of the battery. There, voltage varies as you move between the anode and cathode of a local cell. There is no problem with the equation.
If you would really like to apply it to a wire (which should advisedly have considerable resistance or it will 'short' the battery out), connected to the battery, then probably the closest you can get to applying the equation is to see the conductor ends (at cathode and anode) as circular plates, with a voltage difference between them, causing an electric field. As you move between them, your voltage changes in proportion to your relative distance between the cathode and anode. If you include more complex elements connected along the wire, like resistors, capacitors, inductances, then things are more complicated, and voltage change along the wire will depend on the value, quantity and position of the extra components connected along the wire.
Dlamini
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