Some electrical formulas are simple enough that they can be studied in a straightforward, direct-current circuit, but fundamental and powerful enough to be important stepping stones to studying multi-loop, alternating and semi-conducting circuits. Equations can become significantly complicated by adding just one element, but rules that hold at the basic level go far in taming such complications.
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The statement of resistance for a voltage drop V through a resistor is R=V/I. This is often called Ohm's law. This is a common misconception. Not all resistors obey Ohm's law, but all resistors obey the equation V=IR. A conducting material obeys Ohm's law if its resistivity is independent of the magnitude and direction of the applied electric field. In other words, R is constant as V varies. Graphically, V plotted against I is a straight line.
An exception to Ohm's law is a diode, a simple type of semiconductor whose current is unidirectional and requires a minimum electromotive force (EMF) applied to experience a current. Calculators and computers are full of resistors that do not obey Ohm's law.
Kirchhoff's Junction Rule
Suppose a current I1 enters a circuit junction, and the incoming charges are split into two paths. Call their currents I2 and I3. Then it must hold that I1=I2+I3. This is a conservation of mass rule, or continuity equation. The charge cannot be created or destroyed. If it goes into a junction on one side, all the charges must come out the other side.
Kirchhoff's Loop Rule
As Kirchhoff's junction rule is a result of the law of conservation of mass, his loop rule is a result of the law of conservation of energy. It states that the voltage gains and drops along any closed loop must sum to zero voltage drop. For example, a 12V battery in series with a 200 ohm and 100 ohm resistor would be written as 12V - 200Ω---I - 100Ω---I = 0. In this way, the current can be found. Note that the electrical potential rises across the battery but drops across the resistors.
The rule can be applied to parallel circuits as well, since it holds for multi-loop circuits. It should be noted that subtracting out RI is not a hard rule. If multiple EMF sources are involved in multi-loop circuits, the current may not be in the same direction throughout a loop, and RI would take the opposite sign because it increases the potential instead of decreasing it.
Resistors in Parallel
Resistors in series are merely summed up to determine the effective resistance: Reff = R1 + R2 + .... Resistors in parallel are trickier.
The voltage drop through any closed circuit must equal the battery's voltage. So in the image, the voltage drop across both resistors is the same: V1=V2.
By continuity, the current before and after the junctures must be the same. So if the current out of the battery is I, the currents across the resistors must be such that I = I1 + I2.
The effective resistance of the two resistors in parallel is therefore V=IReff. So Reff = V/(I1+I2). Moving the voltage into the denominator gives the well-known formula Reff = 1/(R1+R2).
Power is the rate that work is done. Work on a charge, e, is eV, since voltage is defined as the energy (work) per charge done by the electric field (or EMF) to move the charge between points of different electrical potential.
Therefore, power = eV/time. But charge passing a point in a measure of time is the current. So P=IV. It is measured in watts (Joules per second).
Since V=IR, P can also be written as I^2R.
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