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  • Draw a circuit with resistors in parallel and in series.
  • Calculate the voltage drop of a current across a resistor using Ohm’s law.
  • Contrast the way total resistance is calculated for resistors in series and in parallel.
  • Explain why total resistance of a parallel circuit is less than the smallest resistance of any of the resistors in that circuit.
  • Calculate total resistance of a circuit that contains a mixture of resistors connected in series and in parallel.

Most circuits have more than one component, called a resistor    that limits the flow of charge in the circuit. A measure of this limit on charge flow is called resistance    . The simplest combinations of resistors are the series and parallel connections illustrated in [link] . The total resistance of a combination of resistors depends on both their individual values and how they are connected.

In part a of the figure, resistors labeled R sub 1, R sub 2, R sub 3, and R sub 4 are connected in series along one path of a circuit. In part b of the figure, the same resistors are connected along parallel paths of a circuit.
(a) A series connection of resistors. (b) A parallel connection of resistors.

Resistors in series

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 R 1 size 12{R rSub { size 8{1} } } {} in [link] (a) could be the resistance of the screwdriver’s shaft, R 2 size 12{R rSub { size 8{2} } } {} the resistance of its handle, R 3 size 12{R rSub { size 8{3} } } {} the person’s body resistance, and R 4 size 12{R rSub { size 8{4} } } {} 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.)

Two electrical circuits are compared. The first one has three resistors, R sub one, R sub two, and R sub three, connected in series with a voltage source V to form a closed circuit. The first circuit is equivalent to the second circuit, which has a single resistor R sub s connected to a voltage source V. Both circuits carry a current I, which starts from the positive end of the voltage source and moves in a clockwise direction around the circuit.
Three resistors connected in series to a battery (left) and the equivalent single or series resistance (right).

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 [link] .

According to Ohm’s law    , the voltage drop, V size 12{V} {} , across a resistor when a current flows through it is calculated using the equation V = IR size 12{V= ital "IR"} {} , where I size 12{I} {} equals the current in amps (A) and R size 12{R} {} is the resistance in ohms Ω size 12{ left ( %OMEGA right )} {} . Another way to think of this is that V size 12{V} {} is the voltage necessary to make a current I size 12{I} {} flow through a resistance R size 12{R} {} .

So the voltage drop across R 1 size 12{R rSub { size 8{1} } } {} is V 1 = IR 1 size 12{V rSub { size 8{1} } = ital "IR" rSub { size 8{1} } } {} , that across R 2 size 12{R rSub { size 8{2} } } {} is V 2 = IR 2 size 12{V rSub { size 8{2} } = ital "IR" rSub { size 8{2} } } {} , and that across R 3 size 12{R rSub { size 8{3} } } {} is V 3 = IR 3 size 12{V rSub { size 8{3} } = ital "IR" rSub { size 8{3} } } {} . The sum of these voltages equals the voltage output of the source; that is,

V = V 1 + V 2 + V 3 . size 12{V=V rSub { size 8{1} } +V rSub { size 8{2} } +V rSub { size 8{3} } } {}

This equation is based on the conservation of energy and conservation of charge. Electrical potential energy can be described by the equation PE = qV size 12{ ital "PE"= ital "qV"} {} , where q size 12{q} {} is the electric charge and V size 12{V} {} is the voltage. Thus the energy supplied by the source is qV size 12{ ital "qV"} {} , while that dissipated by the resistors is

qV 1 + qV 2 + qV 3 . size 12{ ital "qV" rSub { size 8{1} } + ital "qV" rSub { size 8{2} } + ital "qV" rSub { size 8{3} } } {}

Connections: conservation laws

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.

Questions & Answers

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The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
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Practice Key Terms 9

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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