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When are resistors in series? Resistors are in series whenever the flow of charge, called the current, must flow through devices sequentially. For example, if current flows through a person holding a screwdriver and into the Earth, then in [link](a) could be the resistance of the screwdriver’s shaft, the resistance of its handle, the person’s body resistance, and the resistance of her shoes.
[link] shows resistors in series connected to a voltage source. It seems reasonable that the total resistance is the sum of the individual resistances, considering that the current has to pass through each resistor in sequence. (This fact would be an advantage to a person wishing to avoid an electrical shock, who could reduce the current by wearing high-resistance rubber-soled shoes. It could be a disadvantage if one of the resistances were a faulty high-resistance cord to an appliance that would reduce the operating current.)
To verify that resistances in series do indeed add, let us consider the loss of electrical power, called a voltage drop, in each resistor in
According to Ohm’s law, the voltage drop, , across a resistor when a current flows through it is calculated using the equation , where equals the current in amps (A) and is the resistance in ohms . Another way to think of this is that is the voltage necessary to make a current flow through a resistance .
So the voltage drop across is , that across is , and that across is . The sum of these voltages equals the voltage output of the source; that is,
This equation is based on the conservation of energy and conservation of charge. Electrical potential energy can be described by the equation , where is the electric charge and is the voltage. Thus the energy supplied by the source is , while that dissipated by the resistors is
The derivations of the expressions for series and parallel resistance are based on the laws of conservation of energy and conservation of charge, which state that total charge and total energy are constant in any process. These two laws are directly involved in all electrical phenomena and will be invoked repeatedly to explain both specific effects and the general behavior of electricity.
These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus, . The charge cancels, yielding , as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.
RESISTORS IN PARALLEL
[link] shows resistors in parallel, wired to a voltage source. Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it.Each resistor draws the same current it would if it alone were connected to the voltage source (provided the voltage source is not overloaded). For example, an automobile’s headlights, radio, and so on, are wired in parallel, so that they utilize the full voltage of the source and can operate completely independently. The same is true in your house, or any building.
To find an expression for the equivalent parallel resistance , let us consider the currents that flow and how they are related to resistance. Since each resistor in the circuit has the full voltage, the currents flowing through the individual resistors are , , and . Conservation of charge implies that the total current produced by the source is the sum of these currents.
The terms inside the parentheses in the last two equations must be equal. Generalizing to any number of resistors, the total resistance of a parallel connection is related to the individual resistances by
This relationship results in a total resistance that is less than the smallest of the individual resistances. (This is seen in the next example.) When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.
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