Question

Resistors are manufactured so that their resistance lies within a tolerance band. Calculate the maximum and minimum values of the resistances given by: (a) \(10 \Omega \pm 3 \%\) (b) \(29 \mathrm{k} \Omega \pm 5 \%\) (c) \(3 \mathrm{M} \Omega \pm 0.1 \%\)

Short answer

(a) \(10 \Omega \pm 3 \%\) (b) \(29 \mathrm{k} \Omega \pm 5 \%\) (c) \(3 \mathrm{M} \Omega \pm 0.1 \%\) Answer: The minimum and maximum resistance values are as follows: (a) \(9.7 \Omega\) to \(10.3 \Omega\) (b) \(27550 \Omega\) to \(30450 \Omega\) (c) \(2.997 × 10^6 \Omega\) to \(3.003 × 10^6 \Omega\)

Step-by-step solution

(a) Finding minimum and maximum resistance values for \(10 \Omega \pm 3 \%\)

First, find the tolerance value in ohms by multiplying the nominal resistance by the percentage tolerance and dividing by 100: \(Tolerance = 10 \Omega × (3/100) = 0.3 \Omega\) Now, find the minimum and maximum resistance values by subtracting and adding the tolerance value from/to the nominal resistance: Minimum Resistance = \(10 \Omega - 0.3 \Omega = 9.7 \Omega\) Maximum Resistance = \(10 \Omega + 0.3 \Omega = 10.3 \Omega\)

(b) Finding minimum and maximum resistance values for \(29 \mathrm{k} \Omega \pm 5 \%\)

First, convert the nominal resistance to ohms: \(29 \mathrm{k} \Omega = 29000 \Omega\) Next, find the tolerance value in ohms: \(Tolerance = 29000 \Omega × (5/100) = 1450 \Omega\) Now, find the minimum and maximum resistance values: Minimum Resistance = \(29000 \Omega - 1450 \Omega = 27550 \Omega\) Maximum Resistance = \(29000 \Omega + 1450 \Omega = 30450 \Omega\)

(c) Finding minimum and maximum resistance values for \(3 \mathrm{M} \Omega \pm 0.1 \%\)

First, convert the nominal resistance to ohms: \(3 \mathrm{M} \Omega = 3 × 10^6 \Omega\) Next, find the tolerance value in ohms: \(Tolerance = 3 × 10^6 \Omega × (0.1/100) = 3000 \Omega\) Now, find the minimum and maximum resistance values: Minimum Resistance = \(3 × 10^6 \Omega - 3000 \Omega = 2.997 × 10^6 \Omega\) Maximum Resistance = \(3 × 10^6 \Omega + 3000 \Omega = 3.003 × 10^6 \Omega\) So, the maximum and minimum resistance values are as follows: (a) \(9.7 \Omega\) to \(10.3 \Omega\) (b) \(27550 \Omega\) to \(30450 \Omega\) (c) \(2.997 × 10^6 \Omega\) to \(3.003 × 10^6 \Omega\)

Key concepts

Understanding Electrical Resistance

Electrical resistance is a fundamental concept in the field of electronics and electrical engineering. It measures how strongly a material opposes the flow of electric current. The resistance of a conductor is dictated by the material's properties, its length, and cross-sectional area. Engineers often use resistors, components specifically designed to provide a precise resistance to control the flow of current within circuits.

Resistance is measured in ohms (symbol: \( \Omega \)), and it can be calculated using Ohm's Law, which is a cornerstone in understanding electrical circuits. To visualize this, think of water flowing through a pipe: resistance is like the friction that slows the water down. Just as narrower pipes increase friction and reduce water flow, higher resistance decreases current flow in a circuit. By understanding resistance, students can better comprehend how electronic devices control and use electrical energy.

Calculating Tolerance in Resistors

Tolerance percentage is linked to the reliability and precision of resistors. It indicates how much the actual resistance value of the resistor can deviate from its stated value. Manufacturers denote tolerance as a percentage, and you'll often find it marked on the resistor itself. For example, a \( 10 \Omega \pm 3\% \) resistor can have a resistance that's within 3% above or below \( 10 \Omega \).

To compute the range of possible resistance values, multiply the nominal resistance by the tolerance percentage and divide by 100. You'll get a value that you add and subtract from the stated resistance to find the maximum and minimum possible resistances. This range is crucial for designing circuits with precise requirements, as it ensures components operate safely and effectively within expected tolerances.

Ohm's Law – The Core Principle

Ohm's Law is a key principle in electrical engineering that describes the relationship between voltage, current, and resistance in an electric circuit. It states that the current (\( I \)) flowing through a conductor between two points is directly proportional to the voltage (\( V \)) across the two points, and inversely proportional to the resistance (\( R \)) of the conductor. The law is usually expressed in the formula \( V = I \times R \.

Understanding Ohm's Law is crucial for solving electrical problems because it allows us to calculate one of these three quantities (voltage, current, or resistance) if the other two are known. It's particularly relevant to the problem we are examining because once we calculate the minimum and maximum resistances of a resistor (given its tolerance), we can predict how it will behave under different voltages, which is fundamental in circuit design and troubleshooting.