MCAT Physics and Math Review

Chapter 6: Circuits

6.2 Resistance

Resistance is the opposition within any material to the movement and flow of charge. Electrical resistance can be thought of like friction, air resistance, or viscous drag: in all of these cases, motion is being opposed. Materials that offer almost no resistance are called conductors, and those materials that offer very high resistance are called insulators. Conductive materials that offer amounts of resistance between these two extremes are called resistors.

PROPERTIES OF RESISTORS

The resistance of a resistor is dependent upon certain characteristics of the resistor, including resistivity, length, cross-sectional area, and temperature. Three of these are summarized by the equation for resistance:

Equation 6.4

where ρ is the resistivity, L is the length of the resistor, and A is its cross-sectional area. We will explore the effects of each of these variables and temperature in this section.

MCAT EXPERTISE

On the MCAT, the most common resistors you will see outside of generic, unlabeled resistors are light bulbs, although all appliances function as resistors. You may also see resistance applied atypically, as in resistance to air flow in the lungs or to blood moving in the circulatory system. The same mathematical relationships will be useful in both circumstances.

Resistivity

Some materials are intrinsically better conductors of electricity than others. For example, copper conducts electricity better than plastic, which is why electrical wires have a copper core surrounded by a layer of plastic rather than the other way around. The number that characterizes the intrinsic resistance to current flow in a material is called the resistivity (ρ), for which the SI unit is the ohm–meter (Ω · m).

MCAT EXPERTISE

It is always good practice to be able to derive the units of a variable from equations you know. If you were to solve for resistivity (ρ) in the resistance equation, you’d end up with  . Because A is in square meters, R is in ohms, and L is in meters, it simplifies to meters times ohms, which are the units for resistivity.

Length

According to the resistance equation, the resistance of a resistor is directly proportional to its length. A longer resistor means that electrons will have to travel a greater distance through a resistant material. This factor scales linearly: if a resistor doubles its length, it will also double its resistance.

Cross-Sectional Area

The equation for resistance also demonstrates an inverse proportionality between resistance and the cross-sectional area of the resistor: if a resistor’s cross-sectional area is doubled, its resistance will be cut in half. This is because the increase in cross-sectional area increases the number of pathways through the resistor, called conduction pathways. The wider the resistor, the more current that can flow. This is analogous to a river, where the wider the river, the less resistance there is to water flow. Note, however, that electrical current does not follow the continuity equation that applies to incompressible fluids (A1ν1 = A2ν2); it instead obeys Kirchhoff’s laws.

Temperature

Although not evident from the resistance equation, most conductors have greater resistance at higher temperatures. This is due to increased thermal oscillation of the atoms in the conductive material, which produces a greater resistance to electron flow. Because temperature is an intrinsic quality of all matter, we can think of the resistivity as a function of temperature. A few materials do not follow this general rule, including glass, pure silicon, and most semiconductors.

OHM’S LAW AND POWER

Now that we’ve covered voltage, current, and resistance, we can begin to bring these variables together to solve circuits.

Ohm’s Law

Electrical resistance results in an energy loss, which reflects a drop in electrical potential. The voltage drop between any two points in a circuit can be calculated according to Ohm’s law:

V = IR

Equation 6.5

where V is the voltage drop, I is the current, and R is the magnitude of the resistance, measured in ohms (). Ohm’s law is the basic law of electricity because it states that for a given magnitude of resistance, the voltage drop across the resistor will be proportional to the magnitude of the current. Likewise, for a given resistance, the magnitude of the current will be proportional to the magnitude of the emf (voltage) impressed upon the circuit. The equation applies to a single resistor within a circuit, to any part of a circuit, or to an entire circuit (provided one can calculate the equivalent resistance from all of the resistors in the circuit). As current moves through a set of resistors in a circuit, the voltage drops some amount in each resistor; the current (or sum of currents for a divided circuit) is constant. No charge is gained or lost through a resistor; thus, if resistors are connected in series, all of the current must pass through each resistor.

Conductive materials, such as copper wires, act as weak resistors themselves, offering some magnitude of resistance to current and causing a drop in electrical potential (voltage). Even the very sources of emf, such as batteries, have some small but measurable amount of internal resistance,rint. As a result of this internal resistance, the voltage supplied to a circuit is reduced from its theoretical emf value by some small amount. The actual voltage supplied by a cell to a circuit can be calculated from

Ecell – irint

Equation 6.6

where V is the voltage provided by the cell, Ecell is the emf of the cell, i is the current through the cell, and rint is its internal resistance.

MCAT EXPERTISE

Most batteries on the MCAT will be considered “perfect batteries,” and you will not have to account for their internal resistances.

If the cell is not actually driving any current (such as when a switch is in the open position), then the internal resistance is zero, and the voltage of the cell is equal to its emf. For cases when the current is not zero and the internal resistance is not negligible, then voltage will be less than emf.

When a cell is discharging, it supplies current, and the current flows from the positive, higher potential end of the cell around the circuit to the negative, lower potential end. Certain types of cells (called secondary batteries) can be recharged. When these batteries are being recharged, an external voltage is applied in such a way to drive current toward the positive end of the secondary battery. In electrochemical terms, the cell acts as a galvanic (voltaic) cell when it discharges and as an electrolytic cell when it recharges.

REAL WORLD

Superconductors, a special class of materials, are the one major exception to the rules of internal resistance. When these elements and compounds are cooled to very low temperatures (usually well below 100 K, but the exact threshold varies by material), the resistivity of the material (ρ) completely dissipates, dropping to zero. Because these materials break a generally applicable law of physics, superconductors have many interesting and unusual applications.

Measuring Power

In Chapter 2 of MCAT Physics and Math Review, we briefly mentioned that power is the rate at which energy is transferred or transformed. Power is measured as the ratio of work (energy expenditure) to time and can be expressed as follows:

Equation 6.7

In electric circuits, energy is supplied by the cell that houses a spontaneous oxidation–reduction reaction, which when allowed to proceed (by the closing of a switch, for example), generates a flow of electrons. These electrons, which have electrical potential energy, convert that energy into kinetic energy as they move around the circuit, driven by the emf of the cell. As mentioned above, emf is not a force, but is better thought of as a pressure to move, exerted by the cell on the electrons. Current delivers energy to the various resistors, which convert this energy to some other form, depending on the particular configuration of the resistor. One particularly recognizable example of resistors at work is the coils inside a toaster. The coils turn red-hot when the toaster is powered on and dissipate thermal energy, which is a direct consequence of the resistance that the coils pose to the current running through them.

The rate at which energy is dissipated by a resistor is the power of the resistor and can be calculated from

Equation 6.8

where I is the current through the resistor, V is the voltage drop across the resistor, and R is the resistance of the resistor. Note that these different versions of the power equation can be interconverted by substitution using Ohm’s law (V = IR).

MCAT EXPERTISE

These equations for calculating the power of a resistor or collection of resistors are extremely helpful for the MCAT. Commit them to memory—and, more importantly, understand them—and your efforts will be rewarded as points on Test Day.

REAL WORLD

Because power equals voltage times current, power companies can manipulate these two values while keeping power constant. One option is to increase current, which results in a decrease in voltage. The other option would be to increase voltage, thus decreasing the current. Power lines are high-voltage lines, which allows them to carry a smaller current—thus decreasing the amount of energy lost from the system.

RESISTORS IN SERIES AND PARALLEL

Resistors can be connected into a circuit in one of two ways: either in series, in which all current must pass sequentially through each resistor connected in a linear arrangement, or in parallel, in which the current will divide to pass through resistors separately.

Resistors in Series

For resistors connected in series, the current has no choice but to travel through each resistor in order to return to the cell, as shown in Figure 6.1.

Figure 6.1. Resistors in Series Rs increases as more resistors are added.

As the electrons flow through each resistor, energy is dissipated, and there is a voltage drop associated with each resistor. The voltage drops are additive; that is, for a series of resistors, R1R2R3, ⋯ Rn, the total voltage drop will be

Vs = V1 + V2 + V3 + ⋯ + Vn

Equation 6.9

Because V = IR, we can also see that the resistances of resistors in series are also additive, such that

Rs = R1 + R2 + R3 + ⋯ + Rn

Equation 6.10

The set of resistors wired in series can be treated as a single resistor with a resistance equal to the sum of the individual resistances, termed the equivalent or resultant resistance. Note that Rs will always increase as more resistors are added.

KEY CONCEPT

When there is only one path for the current to take, the current will be the same at every point in the line, including through every resistor. Once you know the current of the whole circuit, you can use V = IR to solve for the voltage drop across each resistor (assuming you know the resistances of the resistors).

Example:

A circuit is wired with one cell supplying 5 V in series with three resistors of 3 Ω, 5 Ω, and 7 Ω, also wired in series as shown below. What is the resulting voltage across and current through each resistor of this circuit, as well as the entire circuit?

Solution:

The total resistance of the resistors is

Rs = R1 + R2 + R3 = 3 Ω + 5 Ω + 7 Ω = 15 Ω

Now use Ohm’s law to get the current through the entire circuit:

Because everything is in series, this is also the current through each circuit element. Now, use Ohm’s law for each of the resistors in turn. From a to b, the voltage drop across R1 is

IR1 = (0.33 A)(3 Ω) = 1.0 V

From b to c, the voltage drop across R2 is

IR2 = (0.33 A)(5 Ω) = 1.67 V

From c to d, the voltage drop across R3 is

IR3 = (0.33 A)(7 Ω) = 2.33 V

Resistors in Parallel

Figure 6.2. Resistors in Parallel Rp decreases as more resistors are added.

When resistors are connected in parallel, they are wired with a common high-potential terminal and a common low-potential terminal, as shown in Figure 6.2. This configuration allows charge to follow different parallel paths between the high-potential terminal and the low-potential terminal. In this arrangement, electrons have a “choice” regarding which path they will take: some will choose one pathway, while others will choose a different pathway. No matter which path is taken, however, the voltage drop experienced by each division of current is the same because all pathways originate from a common point and end at a common point within the circuit. This is analogous to a river that splits into multiple streams before plunging over different waterfalls, which then come back together to reform the river at a lower height. If all the water starts at some common height and ends at a lower common height, then it doesn’t matter how many “steps” the water fell over to get to the bottom of the falls: the change in height is the same for each stream. In circuits with parallel arrangements of resistors, this is expressed mathematically as:

Vp = V1 = V2 = V3 = ⋯ = Vn

Equation 6.11

While the voltage is the same for all parallel pathways, the resistance of each pathway may differ. In this case, electrons prefer the path of least resistance; in other words, the current will be largest through the pathways with the lowest resistance. In fact, there is an inverse relationship between the portion of the current that travels through a particular pathway and the resistance offered by that pathway.

KEY CONCEPT

Remember Kirchhoff’s loop rule: if every resistor is in parallel, then the voltage drop across each pathway alone must be equal the voltage of the entire circuit.

The resistance equation previously discussed shows us that there is an inverse relationship between the cross-sectional area of a resistor and the resistance of that resistor. Like opening up rush-hour lanes to reduce traffic congestion or performing cardiac bypass to perfuse hypoxic heart tissue, the configuration of resistors in parallel allows for a greater total number of conduction paths, and the effect of connecting resistors in parallel is a reduction in the equivalent resistance. In effect, we could replace all resistors in parallel with a single resistor that has a resistance that is less than the resistance of the smallest resistor in the circuit. The equivalent resistance of resistors in parallel is calculated by

Equation 6.12

Note that Rp will always decrease as more resistors are added.

Because the voltage drop across any one circuit branch must be same as the voltage drops across each of the other parallel branches, we can see that the magnitude of the current in each branch will be inversely proportional to the resistance offered by each branch. This comes directly from Ohm’s law. Thus, if a circuit divides into two branches and one branch has twice the resistance of the other, the one with twice the resistance will have half the magnitude of current compared to the other. Remember that the sum of the currents going into each division, according to Kirchhoff’s junction rule, must equal the total current going into the point at which the current divides.

Example:

Consider two equal resistors wired in parallel. What is the equivalent resistance of the setup?

Solution:

The equation for summing resistors in parallel is

Next, find the common denominator of the right side:

Then, take the inverse:

This is a special case where R1 = R2. Substituting in, we get:

In the example above, we can see that the total resistance is halved by wiring two identical resistors in parallel. More generally, when n identical resistors are wired in parallel, the total resistance is given by  . Note that the voltage across each of the parallel resistors is equal and that, for equal resistances, the current flowing through each of the resistors is also equal (that is, a current of  runs through each).

Example:

Consider two resistors wired in parallel with R1 = 5 Ω and R2 = 10 Ω. If the voltage across them is 10 V, what is the current through each of the two resistors?

Solution:

First, the current flowing through the whole circuit must be found. To do this, the equivalent resistance must be calculated:

Using Ohm’s law to calculate the current flowing through the circuit gives

Three amps flow through the combination of R1 and R2. Because the resistors are in parallel, Vp = V1 = V2 = 10 V. Apply Ohm’s law to each resistor individually:

As a check, note that Ip = 3 A = I1 + I2. More current flows through the smaller resistor. In particular, note that R1, with half the resistance of R2, has twice the current. Once Ip was found to be 3 A, the problem could have been solved by noting the ratio of the resistances of the two branches.

MCAT EXPERTISE

When approaching circuit problems, the first things you need to find are the total (circuit) values: the total voltage (almost always given as the voltage of the battery), the total (equivalent) resistance, and the total current. To find the total current, first find the total resistance of the circuit.

MCAT Concept Check 6.2:

Before you move on, assess your understanding of the material with these questions.

1.    How does adding or removing a resistor change the total resistance of a circuit with resistors in series? In parallel?

·        Series:

·        Parallel:

2.    What four physical quantities determine the resistance of a resistor?

·         

·         

·         

·         

3.    How does power relate to current, voltage, and resistance?

4.    True or False: The internal resistance of a battery will lower the amount of current it can provide.