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Current refers to the movement of electrical charge from one point to another. Electrons are typically the only carriers of charge in an electric circuit.

Voltage is a bit more difficult to understand in concrete terms. It is related to potential energy. Electrons are attracted to positive charges and will accelerate toward them if free to move. The potential energy of an electron reflects how much kinetic energy it would have if it were accelerated all the way to the positive charge, or the amount of energy it would have taken to move the electron from the positive charge to its current position. Because electric energy is a "conservative field", these quantities are equal. The electrons don't have to move the whole distance to the positive charge, and in electric circuits we are most concerned with "potential difference" between two points partway along the path.

Because electrons have a negative charge, the positive direction of current flow is actually the opposite of the direction of motion of the electrons. Also, negative voltage is where electrons have higher potential energy. Because both of these are reversed, it is rarely necessary to be aware of these inconsistencies in practice.

The basic rules for analyzing a circuit follow from these facts.

Since current is a measure of flow, and electrons are neither created nor destroyed, the current flowing into a point in a circuit must equal the current flowing out.

Because voltage is a conservative field, voltage difference around any complete path through the circuit is 0. As much energy as an electron gains going out, it will lose returning to its original position. Voltage is defined for any particular point in a circuit, regardless of the path taken to that point. (However, voltage can change over time.)

Different electrical components have characteristic relationships between the current through them and the voltage difference between their terminals, and their designed parameters. A battery has one of the simplest relationships. The voltage difference is constant regardless (theoretically) of the current. A resistor is described by the formula V1 - V0 = iR, where the voltage difference (V1 - V0) is proportional to the current (i) and the designed resistance (R) of the resistor.

Connective wires have practically no resistance compared to other components in the circuit. So you can assume the voltage difference across the wire is V1 - V0 = i * 0 = 0. Therefore any terminals connected by unbroken paths of wires can be assumed to have the same voltage.

Define a voltage variable for each such wired node, and a current variable for each component between these nodes. For each voltage node, write an equation adding all currents into the node, and subtracting all nodes out, and placing 0 on the right hand side. Add the characteristic equations for the components. Then solve the system of equations.

It might help intuitive understanding of the equations to correctly guess the directions of the currents, but it is not necessary. The only requirement is that a current added to the equation for one terminal should be subtracted from the equation for the opposite terminal. If you guess the wrong direction for the current, the result will come out negative.

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