OHM: RESISTANCE MEASUREMENTS, EXAMPLES AND SOLVED EXERCISE - PHYSICAL - 2025

The ohm or ohm is the unit of measurement of electrical resistance belonging to the International System of Units (SI), widely used in science and engineering. It was named after the German physicist Georg Simon Ohm (1789-1854).

Ohm was a professor and researcher at the University of Munich, and among his many contributions to electricity and magnetism is the definition of resistance through the relationship between voltage and current through a conductor.

Figure 1. Varied resistors forming part of a circuit. Source: Wikimedia Commons.

This relationship is known as Ohm's Law and is usually expressed as:

R = ΔV / I

Where R represents electrical resistance, ΔV is voltage in volts (V), and I is current in amps (A), all in SI units.

Therefore, 1 ohm, which is also denoted interchangeably by the Greek letter Ω, equals 1 V / A. It means that if setting a voltage of 1 V across a certain conductor causes a current of 1 A, the resistance of that conductor is 1 Ω.

Electrical resistance is a very common circuit element that is used in many ways to properly control current, whether it is part of an integrated circuit or individually.

Measurement of electrical resistance

Figure 5. Georg Simon Ohm, named after the unit for resistance, was born in Bavaria in 1789 and made major contributions to electricity, acoustics, and light wave interference. Source: Wikimedia Commons.

Resistances are measured with the help of a multimeter, a meter that comes in both analog and digital versions. The most basic ones measure direct voltages and currents, but there are more sophisticated devices with additional functions. When used to measure resistance they are called ohmmeters or ohmmeters. This device is very simple to use:

- The central selector is placed in the position to measure resistance, choosing one of the scales identified with the Ω symbol, in case the instrument has more than one.

- The resistance to be measured is extracted from the circuit. If this is not possible, the power supply must be switched off.

- The resistance is placed between the tips or probes of the instrument. Polarity doesn't matter.

- The value is read directly on the digital display. If the instrument is analog, it has a scale marked with the Ω symbol that is read from right to left.

In the following figure (number 2), a digital multimeter and its probes or tips are shown. The model has a single scale for measuring resistance, indicated by an arrow.

Figure 2. Digital multimeter. Source: Pixabay.

The value of a commercial electrical resistance is often expressed by a color band code on its exterior. For example, the resistors in Figure 1 have red, purple, gold, yellow, and gray bands. Each color has a numerical meaning indicating the nominal value, as will be shown below.

Color code for resistors

The following table shows the color codes for the resistors:

Table 1.

Taking into account that the metallic band is on the right, the code is used as follows:

- The first two colors from left to right give the resistance value.

- The third color indicates the power of 10 by which it must be multiplied.

- And the fourth indicates the tolerance established by the manufacturer.

Examples of resistor values

As an example, let's first look at the resistor in the foreground, to the left of figure 1. The sequence of colors shown is: gray, red, red, gold. Remember that the gold or silver band must be on the right.

The gray represents 8, the red is 2, the multiplier is red and equal to 10 ² = 100 and finally, the tolerance is gold which symbolizes 5%. Therefore the resistance is 82 x 100 Ω = 8200 Ω.

Being the tolerance of 5%, it is equivalent in ohms to: 8200 x (5/100) Ω = 410 Ω. Therefore, the resistance value is between: 8200 - 410 Ω = 7790 Ω and 8200 + 410 Ω = 8610 Ω.

Using the color code, you have the nominal or factory value of the resistance, but to make the measurement more precise, you need to measure the resistance with the multimeter, as explained before.

Another example for the resistance of the following figure:

Figure 3. Use of the color code in a resistor R. Source: Wikimedia Commons.

We have the following for resistor R: red (= 2), violet (= 7), green (multiply by 10 ⁵), so the resistor R in the figure is 27 x 10 ⁵ Ω. The tolerance band is silver: 27 x 10 ⁵ x (10/100) Ω = 27 x 10 ⁴ Ω. One way to express the above result, rounding 27 x 10 ⁴ to 30 x 10 ⁴, is:

Most used prefixes

The values ​​that an electrical resistance can have, which is always positive, are in a very wide range. For this reason, powers of 10 are widely used to express their values, as well as prefixes. Here are the most common:

Table 2.

According to this notation, the resistance in the previous example is: (2.7 ± 0.3) MΩ.

Resistance of a conductor

The resistors are made of different materials and it is a measure of the opposition that the conductor has to the passage of current, as is known, not all materials conduct in the same way. Even between materials considered as conductors there are differences.

Resistance depends on several characteristics, the most important being:

- Geometry of the conductor: length and area of ​​the cross section.

- Resistivity of the material: indicates the opposition that the material presents to the passage of current.

- Temperature: resistivity and resistance increase with temperature, since the internal ordering of the material decreases and thus the current carriers are hindered in their passage.

For a conductor of constant cross section, at a given temperature the resistance is given by:

R = ρ (ℓ / A)

Where ρ is the resistivity of the material at the temperature in question, which is determined experimentally, ℓ is the length of the conductor and A is the cross-sectional area.

Figure 4. Resistance of a conductor. Source: Wikimedia Commons.

Exercise resolved

Find the resistance of a copper wire of radius 0.32 mm and 15 cm long, knowing that the resistivity of copper is 1.7 × 10 ^-8 Ω.m.

Solution

Given that the resistivity is in units of the International System, the most appropriate thing is to express the cross-sectional area and the length in these units, and then substitute in the formula of the preceding section:

Radius = 0.32 mm = 0.32 × 10 ^-3 m

A = π (Radius ²) = π (0.32 × 10 ^-3 m) ² = 3.22 x 10 ^-7 m ²

ℓ = 15 cm = 15 x 10 ^-2 m

R = ρ (ℓ / A) = 1.7 × 10 ^-8 Ω.mx (15 x 10 ^-2 m / 3.22 x 10 ^-7 m ²) = 7.9 × 10 ^-3 Ω = 7.9 m-ohm.

References

1. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 5. Electrostatics. Edited by Douglas Figueroa (USB).
2. Giancoli, D. 2006. Physics: Principles with Applications. 6 ^th. Ed Prentice Hall.
3. Resnick, R. (1999). Physical. Vol. 2. 3 ^rd in Spanish. Compañía Editorial Continental SA de CV
4. Sears, Zemansky. 2016. University Physics with Modern Physics. 14 ^th. Ed. Volume 2.
5. Serway, R., Jewett, J. (2018). Physics for Science and Engineering. Volume 1. 10 ^ma. Ed. Cengage Learning.