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Chapter 3: Problem 39

Use a variation model to solve for the unknown value. The current in a wire varies directly as the voltage and inversely as the resistance. If the current is 9 amperes (A) when the voltage is 90 volts \((\mathrm{V})\) and the resistance is 10 ohms \((\Omega)\), find the current when the voltage is \(160 \mathrm{~V}\) and the resistance is \(5 \Omega\).

Short Answer

Expert verified
The current is 32 A when the voltage is 160 V and the resistance is 5 惟.

Step by step solution

01

Understand the Variation Model

The current (I) varies directly as the voltage (V) and inversely as the resistance (R). This relationship can be represented by the formula: \[ I = k \frac{V}{R} \] where k is the constant of variation.
02

Determine the Constant of Variation (k)

Use the given values to find the constant k. When the current (I) is 9 A, the voltage (V) is 90 V, and the resistance (R) is 10 惟.Plug these values into the formula: \[ 9 = k \frac{90}{10} \]Solve for k: \[ 9 = k \times 9 \]\[ k = 1 \]
03

Set Up the Equation with the New Values

Use the determined constant k to find the current (I) when the voltage (V) is 160 V and the resistance (R) is 5 惟. Plug the values into the formula: \[ I = 1 \frac{160}{5} \]
04

Solve for the New Current

Simplify the equation to find the current: \[ I = 1 \times \frac{160}{5} = 32 \]
05

State the Final Answer

The current when the voltage is 160 V and the resistance is 5 惟 is 32 A.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

current-voltage relationship
In electrical circuits, understanding the relationship between current and voltage is crucial. The current flowing through a conductor depends on the voltage applied across it. This is because voltage acts as a driving force that pushes the electrons through the conductor.
When voltage increases, the electric current also increases since there is more force to drive the electrons. Conversely, when voltage decreases, the current also decreases.
It's essential to recognize that this relationship can be influenced by other factors such as resistance. By grasping this concept, you can predict how changes in voltage will affect the current in a given circuit.
ohm's law
Ohm's Law is fundamental in understanding electrical circuits. It 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) of the conductor.
The formula is represented as: \[ I = \frac{V}{R} \]
If you know two of the three variables (current, voltage, and resistance), you can easily calculate the third by rearranging the formula. For example, if you want to find the voltage, the formula becomes: \[ V = I \times R \]
Ohm's Law is a powerful tool in both theoretical calculations and practical applications, such as troubleshooting electrical problems.
variation constant
A variation constant (k) is used in equations to link variables exhibiting direct or inverse proportionality. In the context of the current-voltage relationship, the current (I) varies directly with the voltage (V) and inversely with the resistance (R).
The general formula can be expressed as: \[ I = k \frac{V}{R} \]
To find the variation constant, you need known values of current, voltage, and resistance. For instance, using the given values (I = 9 A, V = 90 V, R = 10 惟):\[ 9 = k \frac{90}{10} \]
Solve for k:\[ 9 = k \times 9 \] \[ k = 1 \]
Once k is determined, it can be used to find the current for different values of voltage and resistance, making it a powerful tool for solving electrical problems.

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