- How is it calculated?
- The resistance of a conductor
- Examples
- Experiment to check Ohm's law
- Hydraulic analogy of Ohm's law
- Resistors and switches
- Exercises
- - Exercise 1
- Solution
- - Exercise 2
- Solution
- References
The Ohm 's law in its macroscopic form, indicates that the voltage and intensity of current in a circuit is directly proportional resistance being the constant of proportionality. Denoting these three quantities as V, I and R respectively, Ohm's law states that: V = IR
Likewise, Ohm's law is generalized to include circuit elements that are not purely resistive in alternating current circuits, in this way it takes the following form: V = IZ
Figure 1. Ohm's law is applicable to many circuits. Source: Wikimedia Commons. Tlapicka
Where Z is the impedance, which also represents the opposition to the passage of alternating current by a circuit element, for example a capacitor or an inductance.
It should be noted that not all circuit materials and elements comply with Ohm's law. Those in which it is valid are called ohmic elements, and in which it is not fulfilled, they are called non-ohmic or non-linear.
Common electrical resistors are of the ohmic type, but diodes and transistors are not, since the relationship between voltage and current is not linear in them.
Ohm's law owes its name to the Bavarian-born German physicist and mathematician George Simon Ohm (1789-1854), who during his career devoted himself to studying the behavior of electrical circuits. The unit for electrical resistance in the SI International System has been named in his honor: the ohm, which is also expressed by the Greek letter Ω.
How is it calculated?
Although the macroscopic form of Ohm's law is the best known, since it links quantities that are easily measurable in the laboratory, the microscopic form relates two important vector quantities: the electric field E and the current density J:
Where σ is the electrical conductivity of the material, a property that indicates how easy it is to conduct current. For its part, J is a vector whose magnitude is the quotient between the intensity of current I and the cross-sectional area A through which it circulates.
It is logical to assume that there is a natural connection between the electric field inside a material and the electric current that circulates through it, such that the greater the current, the more current.
But the current is not a vector, since it does not have a direction in space. On the other hand, the vector J is perpendicular -or normal- to the cross-sectional area of the conductor and its direction is that of the current.
From this form of Ohm's law we arrive at the first equation, assuming a conductor of length ℓ and cross section A, and substituting the magnitudes of J and E by:
The inverse of conductivity is called resistivity and is denoted by the Greek letter ρ:
Thus:
The resistance of a conductor
In the equation V = (ρℓ / A).I, the constant (ρℓ / A) is the resistance, therefore:
The resistance of the conductor depends on three factors:
-Its resistivity ρ, typical of the material with which it is manufactured.
-Length ℓ.
-The area A of its cross section.
The higher ℓ, the greater the resistance, since current carriers have more opportunities to collide with other particles inside the conductor and lose energy. And conversely, the higher A, the easier it is for current carriers to move in an orderly manner through the material.
Finally, in the molecular structure of each material lies the ease with which a substance allows electric current to pass. Thus, for example, metals such as copper, gold, silver and platinum, with low resistivity, are good conductors, while wood, rubber and oil are not, which is why they have higher resistivity.
Examples
Here are two illustrative examples of Ohm's law.
Experiment to check Ohm's law
A simple experience illustrates Ohm's law, for this you need a piece of conductive material, a variable voltage source and a multimeter.
A voltage V is established between the ends of the conductive material, which must be varied little by little. With the variable power source, the values of said voltage can be set, which are measured with the multimeter, as well as the current I that flows through the conductor.
The pairs of V and I values are recorded in a table and with them a graph is constructed on graph paper. If the resulting curve is a straight line, the material is ohmic, but if it is any other curve, the material is non-ohmic.
In the first case, the slope of the line can be determined, which is equivalent to the resistance R of the conductor or to its inverse, the conductance.
In the image below, the blue line represents one of these graphs for an ohmic material. Meanwhile, the yellow and red curves are made of non-ohmic materials, like a semiconductor, for example.
Figure 2. Graph I vs. V for ohmic materials (blue line) and non-ohmic materials. Source: Wikimedia Commons.
Hydraulic analogy of Ohm's law
It is interesting to know that the electric current in Ohm's law has a behavior similar to that of water circulating through a pipe. The English physicist Oliver Lodge was the first to propose the simulation of the behavior of the current using elements of hydraulics.
For example, the pipes represent the conductors, since the water circulates through them and the current carriers through the latter. When there is a constriction in the pipe, the passage of water is difficult, so this would be equivalent to an electrical resistance.
The difference in pressure at two ends of the tube allows the water to flow, which provides a difference in height or a water pump, and similarly, the difference in potential (the battery) is what keeps the charge moving., equivalent to the flow or volume of water per unit of time.
A piston pump would play the role of an alternating voltage source, but the advantage of putting in a water pump is that the hydraulic circuit would thus be closed, just as an electrical circuit must be for current to flow.
Figure 3. Hydraulic analogy for Ohm's law: in a) a water flow system and in b) a simple resistive circuit. Source: Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. McGraw Hill.
Resistors and switches
The equivalent of a switch in a circuit, it would be a stopcock. It is interpreted in this way: if the circuit is open (stopcock closed), the current, like the water, cannot flow.
On the other hand, with the switch closed (stopcock fully open) both current and water can flow without problems through the conductor or pipe.
The stopcock or valve can also represent a resistance: when the tap is fully opened it is equivalent to having a zero resistance or a short circuit. If it closes completely it is like having the circuit open, while partially closed it is like having a resistance of a certain value (see figure 3).
Exercises
- Exercise 1
An electric iron is known to require 2A at 120V to function properly. What is its resistance?
Solution
Solve for resistance from Ohm's law:
- Exercise 2
A wire 3 mm in diameter and 150 m long has an electrical resistance of 3.00 Ω at 20 ° C. Find the resistivity of the material.
Solution
The equation R = ρℓ / A is appropriate, therefore the cross-sectional area needs to be found first:
Finally, when substituting, you get:
References
- Resnick, R. 1992. Physics. Third expanded edition in Spanish. Volume 2. Compañía Editorial Continental SA de CV
- Sears, Zemansky. 2016. University Physics with Modern Physics. 14 th. Ed. Volume 2. 817-820.
- Serway, R., Jewett, J. 2009. Physics for Science and Engineering with Modern Physics. 7th Edition. Volume 2. Cengage Learning. 752-775.
- Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. McGraw Hill.
- Sevilla University. Department of Applied Physics III. Density and intensity of current. Recovered from: us.es.
- Walker, J. 2008. Physics. 4th Ed. Pearson. 725-728