Question

The current in a resistor is inversely proportional to the resistance. By what factor will the current change if a resistance increases \(10.0 \%\) due to heating?

Short answer

The current will decrease by a factor of 1/1.10, or approximately 0.909.

Step-by-step solution

Understand the Proportionality Relation

The current (I) in a resistor is inversely proportional to the resistance (R). This means that when resistance increases, current decreases, and vice versa, following the formula: I = k/R, where k is a constant.

Calculate the New Resistance

If the resistance increases by 10%, the new resistance, R', can be calculated as R' = R + 0.10R = 1.10R.

Find the Factor of Change in Current

Using the formula I = k/R, the new current I' with the increased resistance becomes I' = k/R'. Substituting R' with 1.10R gives I' = k/(1.10R). To find the factor of change, we calculate the ratio of the new current I' to the original current I which is (I'/I) = (k/R) / (k/(1.10R)) = 1/1.10.

Key concepts

Ohm's Law

Ohm's Law is a fundamental concept in the study of electricity, essential for anyone delving into the field of electronics or electrical engineering. It describes the relationship between voltage, current, and resistance in an electric circuit. Mathematically expressed as V = IR, where V is the voltage across the circuit elements, I is the current flowing through the circuit, and R is the resistance.

Simply put, the law states that the current through a conductor between two points is directly proportional to the voltage across the two points, provided the temperature remains constant. The resistance, on the other hand, acts as a limiting factor to the flow of electrons; the higher the resistance, the lower the current for a given voltage. Understanding Ohm's Law is critical as it provides the groundwork for analyzing electric circuits and predicting how they behave under various conditions.

Electrical Resistance

Electrical resistance is the measure of the difficulty electrons face when flowing through a material. It is one of the critical parameters that dictate how well a material can conduct electricity; materials with low resistance are good conductors, while those with high resistance are poor conductors, known as insulators. Resistance is measured in ohms (symbol: \( \Omega \)).

The resistance of a material can vary with several factors such as temperature, material composition, and physical dimensions. For instance, when a resistor heats up due to an electric current, its resistance tends to increase—a phenomenon commonly observed in various electronic components. This change in resistance with temperature must be taken into account to ensure the effective functioning of electrical circuits, particularly when precision is required.

Proportionality in Physics

Proportionality is a concept in physics that describes the relationship between two quantities that change together in a predictable way. When two variables are directly proportional, an increase in one causes a proportional increase in the other, and vice versa. Conversely, inverse proportionality means that as one quantity increases, the other decreases at a proportional rate.

This concept is pervasive in various physical laws and equations, serving as a critical tool for analysis and prediction. In our context, the focus is on inverse proportionality, which is seen in the relationship between current and resistance within an electrical circuit. By appreciating the concept of proportionality, one can better understand and predict how changes in one element of a system will influence the others.

Current Flow in Resistors

The current flow in resistors is dictated by the resistance they offer to the passage of electric charge. When a potential difference or voltage is applied across a resistor, it generates an electric current. However, the greater the resistance, the more it opposes the flow of current. This relationship is inverse, as highlighted in the original exercise, meaning that any increase in resistance leads to a decrease in current, according to the formula \(I = k/R\), where \(k\) is a proportionality constant.

In practical scenarios like a resistor heating up and its resistance changing, as in the textbook exercise, this relationship helps predict the new current flowing through the circuit. By understanding the mathematical representation and physical implications of these concepts, students can solve complex circuit problems effectively.