Question

The formula for electric power, \(P\), is \(\mathrm{P}=1 \times V\), where \(\mathrm{I}\) is the current and \(V\) is the voltage. The formula for voltage is \(\mathrm{V}=\mathrm{I} \times R\), where \(\mathrm{I}\) is also the current and \(R\) is the resistance. How will the power of a given current be affected if the resistance is doubled and the voltage is quadrupled? (A) The power will be doubled. (B) The power will be quadrupled. (C) The power will be 8 times greater. (D) The power will be 16 times greater.

Short answer

The power will be 8 times greater.

Step-by-step solution

Identify the Original Power Formula

We start with the formula for electric power given by \( P = I \times V \), where \( I \) is the current and \( V \) is the voltage.

Substitute Voltage Formula

The formula for voltage \( V \) is \( V = I \times R \). Substitute this into the power formula to get \( P = I \times (I \times R) = I^2 \times R \).

Doubling the Resistance

If the resistance \( R \) is doubled, the new resistance is \( 2R \). Substituting into the power formula gives \( P_{new1} = I^2 \times 2R = 2(I^2 \times R) \). This shows that power doubles with respect to the original resistance.

Quadrupling the Voltage

If the voltage \( V \) is quadrupled, the new voltage is \( 4V \). Continuing from the step where voltage is expressed as \( V = I \times R \), quadrupling gives \( V_{new} = 4(I \times R) \).

Calculate New Power with Adjusted Resistance and Voltage

Substitute the quadrupled voltage into the power equation: \( P_{new2} = I \times 4(I \times R) = 4(I^2 \times R) \).

Combine Resistance and Voltage Changes

When both the resistance is doubled and the voltage quadrupled, the increase in power is cumulative. Substitute the quadrupled power result into the doubled resistance result: \( P_{final} = 2(I^2 \times R) + 4(I^2 \times R) = 8(I^2 \times R) \). This demonstrates that power increases by a factor of 8 when both alterations are applied.

Key concepts

Current and Voltage Relationship

To understand the relationship between current and voltage, we need to consider how these two elements work together within an electric circuit. Current, denoted by the symbol \(I\), is the flow of electric charge. It is measured in amperes (A). Voltage, represented as \(V\), is the electrical potential difference between two points in a circuit and is measured in volts (V). These two are inherently linked through Ohm's Law.
Ohm's Law is a fundamental principle that describes the relationship between current, voltage, and resistance in an electrical circuit. This relationship can be expressed with the formula \(V = I \times R\), where \(R\) is the resistance measured in ohms (Ω). It shows that voltage is directly proportional to both current and resistance.
Understanding this relationship helps in predicting how changes in voltage or resistance can affect current. For example, increasing voltage while keeping resistance constant will increase the current. This principle lays the groundwork for analyzing more complex scenarios in electrical power.

Electric Circuit Resistance

Resistance is a critical component of any electric circuit. It acts as a barrier to the flow of current, affecting how easily electricity can pass through a conductor. Resistance, represented by \(R\), is determined by factors such as the material of the conductor, its diameter, and length.
By manipulating the resistance of a circuit, we can control the amount of current flowing through it. The effect of resistance is so significant that it's an essential consideration when calculating power in electrical systems. For instance, when resistance doubles, as seen in many practical problems, it directly impacts the calculated power by changing the voltage relationship \(V = I \times R\).
In our case study, when the resistance is doubled, it had a direct influence on the power equation. This shows the importance of accurately accounting for resistance in any power calculations. This concept demonstrates how intricate and interlinked electrical calculations require a solid understanding of roles that resistance plays within a circuit.

Power Formula Application

Applying the power formula accurately in different scenarios is crucial for understanding how electric power behaves under changing conditions. The basic power formula \(P = I \times V\) gives a straightforward means to calculate the power consumed or produced in an electric circuit. Power, measured in watts (W), is a measure of how fast energy is being used or generated.
In our example, alterations to the circuit's components like voltage and resistance lead to a deeper analysis using an extended form of the power formula, \(P = I^2 \times R\). This derivative of the power formula helps understand how power changes when both voltage and resistance are modified.
In cases where the voltage is quadrupled and the resistance is doubled, we must consider both effects on the power equation. Each change individually affects power, but combined, they result in an eightfold increase in power usage, as shown by the calculation steps provided. Such exercises illustrate how changes in circuit components interact to affect power, entailing the necessity for precise calculations in complex electric systems.