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When we connect two resistors connected in parallel with a battery, it creates an electric field through this wire. well, my question here: Is the electric field in the main wire is the same as the two wires that are branched from the main? If so why the sum of the current in the two branched wires is the same as the current of the main wire?

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    – DanielSank
    Commented Dec 29, 2015 at 13:38

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You've asked two questions in one. First, no: the electric field through the main wire is NOT the same as that through the two resistors. The electric field is a continuously varying field, and isn't necessarily the same at any two points in a circuit.

Second, you're overthinking this. It's seldom useful to consider the electric field when calculating current through a circuit. Using good old $V=IR$ will get you there in 99% of the cases.

On your specific second question, a current is a stream of electrons moving through a circuit element. If you have a single-loop circuit, then the number of electrons (the current) passing any one point in the circuit will be the same as at any other points. If your circuit splits, then the sum of the electrons (and thus the current) passing through each branch will equal the electrons (and thus the current) through the main circuit.

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The electric field in an ideal wire is always zero.

To calculate the average electric field through a circuit element you take the voltage across the element and divide it by the length of the element. Since an ideal wire has the same voltage at both ends the voltage across the wire is zero and therefore the E field is also zero.

The voltage across a wire is independent of the current through the wire (it is always zero ideally). So the fact the current is different is irrelevant to the voltage or E field.

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