Resistivity

Claimed by Brian Duffy -- Fall 2016 Resistivity is the measure of a specific materials ability to impede the flow of an electric current. The SI unit of resistivity is often measured in [math]\displaystyle{ ({Ohm}⋅{Meter}) }[/math] or [math]\displaystyle{ ({Ω}⋅{m}) }[/math] and is used to determine the resistance of a given conductor. Resistivity of an object is almost entirely dependent on two specific factors: temperature and material. Notably, resistivity, unlike resistance, is independent of the size or shape of a material.

The Main Idea

Resistivity is essentially a constant that describes the resistability of a specific material with respect to the current that passes through it. Some materials will more readily allow the flow of current in comparison to others. For instance, copper has half the resistivity as that of aluminum. Thus, most wires are made out of copper instead of aluminum -- as aluminum impedes the flow of electrical current.

A Mathematical Model

Resistance is often calculate from resistivity using the following equation [math]\displaystyle{ R = \frac{\rho L}{A} }[/math] where R is the resistance [math]\displaystyle{ \rho }[/math] is the resistivity, L is the length, and A is the cross-sectional area.

In a circuit the Electrical Resistance is then used to calculate the current in a circuit using the following equation Ohm's law [math]\displaystyle{ I = \frac{|\Delta V|}{R} }[/math] where V is the voltage, I is the current, and R is the resistance. In this equations voltage and resistance are independent variables, whereas the Current is the dependent variable. This law, while useful, only works for ohmic resistors.

The definition provided above is specific to ohmic resistors, as stated. These resistors have a uniform cross-section, where current flows uniformly through them. Instead, a more general definition starts with the idea that an electric field inside a specific material is responsible for the electric current flowing within it. Thus, the electrical resistivity, or "p" can be defined as the ratio of the electric field to the density of the current it creates:

[math]\displaystyle{ \rho=\frac{E}{J}, \,\! }[/math]

where "ρ" is the resistivity (ohm⋅meter), "E" is the magnitude of the electric field (volts per meter), and J is the magnitude of the current density (amperes per square meter).

Note, when E and J are inside the conductor. Conductivity is the inverse of resistivity:

[math]\displaystyle{ \sigma=\frac{1}{\rho} = \frac{J}{E}. \,\! }[/math]

Water Analogy

The relationship between resistivity and resistance can be thought of as a series of pipes. Electrical Resistance in a particular material is similar to an analogy of pipes of varying diameter. The larger the pipe the easier it is for water to get through. The resistivity of the "pipes" never change, but the cross sectional area does, in order for the facilitation of "water" flow (current).

Resistivity of Materials

Every conductor has a natural resistivity that is relatively consistent at a given temperature. This number is calculated through experimentation. Here is a list of common conductors and their resistivities.

(http://avstop.com/ac/Aviation_Maintenance_Technician_Handbook_General/images/fig10-41.jpg)

Temperature

In addition to each material having a different resistivity. The same material at different temperatures may exhibit different resistivities. As materials heat up they become less facilitative of current flow. This is due to the fact that nuclei are moving faster at a sub-atomic level, making it difficult for electrons to move through.

Examples

3 examples of potential problems involving resistivity and resistance.

Simple

Question

An unknown ohmic resistor is attached to a 3V battery and the current is measured at 1 amp. Calculate the resistance of the unknown resistor.

Answer

Using the equation I=|dV|/R we can substitute is 1 for I and 3 for dV leaving us with the equation 1=3/R. Solving for R we come to the answer that the it must be a 3 ohm resistor.

Middling

Question

A cylinder of an unknown material has a resistance of 30 ohms. Another cylinder made of the exact same material is twice as long and has a radius that is twice as large. What is the resistance of this cylinder?

Answer

Given the equation [math]\displaystyle{ R = \frac{\rho L}{A} }[/math] we know that when the length is doubled the resistance must also double. In addition we know that when the radius is doubled, the cross section area must go up by a factor of 4. This means that the resistance would go down by a factor of 1/4. Putting both of those facts together know that R2 = R1 * 2 * 1/4 or R2 = 15 ohms.

Difficult

Question

A battery and resistor circuit is connected to a very sensitive ohmmeter and is taken outside and left in the sun on a very hot day. What, if anything, will happen to its reading after being outside for a few minutes and why? Assume the battery is unaffected.

Answer

The current would be less that it was inside. Since the circuit was taken outside the resistor would heat up due to the sun. This would in turn cause its resistance to go up. When the resistance goes up and the voltage of the battery stays the same. due to Ohms Law the current must go down, resulting in a lower reading.

Scope

Resistivity is very important to electrical engineers and others who work with circuits because it is important to understand how a circuit is going to work when outside of laboratory conditions. Users cannot expect their circuits to work the same in both nominal and extreme temperature conditions.

2. http://www.britannica.com/technology/resistance-electronics

3. http://www.nist.gov/data/PDFfiles/jpcrd155.pdf

4. http://www.regentsprep.org/Regents/physics/phys03/bresist/default.htm

5. http://forums.extremeoverclocking.com/showthread.php?p=4144637

Resistivity

Contents

The Main Idea

A Mathematical Model

Water Analogy

Resistivity of Materials

Temperature

Examples

Simple

Question

Answer

Middling

Question

Answer

Difficult

Question

Answer

Scope

See also

Further reading

External links

References

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