Question

Applications When a current \(I\) flows through a resistance \(R,\) the power in the resistance is \(I^{2} R .\) If the current is increased by 2.50 amperes, write an expression for the new power, and multiply out.

Short answer

The new power expression after the current is increased by 2.50 amperes is \( P_{new} = (I^2 + 5I + 6.25) R \).

Step-by-step solution

Identify the original power expression

The original power expression is given by the formula for power through a resistance, which is the product of the square of the current (I) and the resistance (R), expressed as \( P = I^2 R \) .

Express the increased current

If the current is increased by 2.50 amperes, the new current is expressed as \( I_{new} = I + 2.50 \) amperes.

Write the new power expression

Using the expression for power and substituting the increased current, the new power \( P_{new} \) can be written as \( P_{new} = I_{new}^2 R \).

Substitute the increased current into the new power expression

Substituting the increased current into the new power expression, we get \( P_{new} = (I + 2.50)^2 R \) .

Expand the new power expression

Expand the squared term in the new power expression using the binomial expansion to get \( P_{new} = (I^2 + 2 \times I \times 2.50 + 2.50^2) R \).

Key concepts

Electrical Current

Electrical current, often simply called current, is a flow of electric charge. In the context of a circuit, it refers to the movement of electrons through a conductor. The standard unit for current is the ampere, symbolized as 'A'. Current is a fundamental parameter in analyzing electrical circuits, as it directly influences other essential variables.

For instance, in the exercise provided, the initial current is denoted as 'I', which then is increased by a specific amount, 2.50 amperes. Understanding how this change in current affects the power across a resistance is crucial in designing and evaluating electrical systems. A higher current typically means more energy is being transferred through the circuit per unit of time, leading to increased power dissipation in forms such as heat.

Resistance

In electrical terms, resistance is a measure of the opposition to current flow within an electrical circuit. Resistance is quantified in ohms (Ω). It plays a critical role in determining how much current flows for a given voltage – as per Ohm's Law, which states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points, and inversely proportional to the resistance (R).

The formula \( I = \frac{V}{R} \) offers a fundamental understanding of this relationship. In our textbook exercise scenario, resistance remains constant even when the current is increased. This consistency allows us to calculate how changes in current affect the electrical power in the resistance.

Power Expression

The 'Power Expression' deals with calculating the rate at which energy is consumed or generated within an electrical component. Power in a resistive component, like a heater or a resistor, is given by \( P = I^2 R \), where 'P' is the power in watts, 'I' represents the current in amperes, and 'R' is the resistance in ohms.

This equation comes from the combination of Ohm's Law and the definition of electrical power, which can also be expressed as \( P = VI \). It indicates that power is directly proportional to the square of the current flowing through the resistance. As such, even small increases in current can result in significant increases in power, as is explored in the exercise.

Binomial Expansion

The 'Binomial Expansion' is a mathematical technique used to expand expressions that are raised to a power. It is based on the binomial theorem, which gives a formula for expressing the expansion of powers of a binomial, such as \( (a+b)^n \).

In our electrical power problem, the binomial expansion is used to simplify the expression \( (I + 2.50)^2 \) into \( I^2 + 2 \times I \times 2.50 + 2.50^2 \). This simplification is crucial for creating an expanded expression that can be easily evaluated for different values of current, 'I'. By understanding the binomial expansion, students can tackle a wide variety of problems that involve powers of expressions.