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Chapter 2: Problem 8

Ohm's law says that when electric current is flowing across a resistor, the current \(i\), measured in amperes, can be calculated from the voltage \(v\), measured in volts, and the resistance \(R\), measured in ohms. The relationship is given by $$ i=\frac{v}{R} \text { amperes. } $$ a. A resistor in a radio circuit is rated at \(4000 \mathrm{ohms}\). i. Find a formula for the current as a function of the voltage. ii. Plot the graph of \(i\) versus \(v\). Include values of the voltage up to 12 volts. iii. What happens to the current when voltage increases? b. The lights on your car operate on a 12 -volt battery. i. Find a formula for the current in your car lights as a function of the resistance. ii. Plot the graph of \(i\) versus \(R\). We suggest a horizontal span here of 1 to \(25 .\) iii. What happens to the current when resistance increases?

Short Answer

Expert verified
a(i) Current increases linearly with voltage. b(iii) Current decreases as resistance increases.

Step by step solution

01

Derive Current as a Function of Voltage

According to Ohm's law, the current through a resistor can be expressed as \( i = \frac{v}{R} \). Given that for the radio circuit, the resistance \( R \) is 4000 ohms, we can substitute this value into the equation to find the current as a function of voltage: \( i(v) = \frac{v}{4000} \) amperes.
02

Plot Current vs Voltage

We will plot the graph of \( i = \frac{v}{4000} \) for voltages \( v \) from 0 to 12 volts. This is a linear graph with a slope of \( \frac{1}{4000} \), indicating that current increases linearly with voltage. A graphing tool can be used to visualize this line, and some example points would be \( (0,0) \), \( (6,0.0015) \), and \( (12,0.003) \).
03

Analyze Current with Increasing Voltage

From the equation \( i = \frac{v}{4000} \), as \( v \) increases, the current \( i \) also increases linearly. This means that as the voltage increases, the current increases proportionally.
04

Derive Current as a Function of Resistance

For car lights operating on a 12-volt battery, we use Ohm's law again: \( i = \frac{v}{R} \). By substituting \( v = 12 \) volts, the equation becomes \( i(R) = \frac{12}{R} \) amperes. This equation expresses current as a function of resistance.
05

Plot Current vs Resistance

We will plot the graph of \( i = \frac{12}{R} \) for resistance \( R \) ranging from 1 to 25 ohms. This graph shows that current decreases as resistance increases, a hyperbolic curve. Example points are \( (1,12) \), \( (5,2.4) \), and \( (25,0.48) \).
06

Analyze Current with Increasing Resistance

Using the equation \( i = \frac{12}{R} \), as resistance \( R \) increases, the current \( i \) decreases non-linearly. This inverse relationship implies that increasing resistance leads to a decrease in current.

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Key Concepts

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

Electric Current
Electric current refers to the flow of electric charge through a conductor, typically measured in amperes (A). It is a fundamental concept in physics and electronics, representing how electricity is conveyed across a circuit.
In Ohm's Law, electric current is directly proportional to voltage and inversely proportional to resistance, illustrated by the formula \( i = \frac{v}{R} \). Here, \( i \) stands for the current, \( v \) the voltage, and \( R \) the resistance.
  • Current increases with a rise in voltage, assuming resistance stays constant.
  • Current decreases as resistance increases, provided voltage remains unchanged.
Thus, controlling current involves adjusting either the voltage or resistance in a circuit. Understanding how these relationships function is key to manipulating electric circuits effectively.
Resistor
A resistor is an essential component of electrical circuits that limits or regulates the flow of electric current. This regulation is important to prevent damage from excessive current or to achieve desired electrical characteristics within the circuit.
In any given circuit, resistors are measured in ohms (惟), which quantify their resistance. For instance, when a resistor is described as having 4000 ohms in a radio circuit, it means that it provides a level of resistance designed to manage the current flowing through it at a specified voltage.
  • Resistors help in balancing current levels.
  • They allow for the voltage to influence current without causing overcurrent conditions.
This component is crucial for the stable operation of electronic devices, ensuring that energy inputs are appropriately controlled.
Voltage
Voltage, often referred to as electric potential difference, represents the force that pushes electric current through a circuit. It is measured in volts (V) and is a critical determinant of how much energy per charge is transferred across circuit elements.
When you think about Ohm鈥檚 Law, voltage is one side of the equation that affects both the current and situation across a resistor by the formula \( i = \frac{v}{R} \).
  • When voltage increases in a circuit with constant resistance, current increases proportionally.
  • A constant voltage across a variable resistor will yield different levels of current.
Understanding voltage helps in determining how much current will flow in a circuit, making it fundamental to designing electrical systems.
Resistance
Resistance is the opposition to the flow of electric current, measured in ohms (惟). It is an intrinsic property of materials that indicates how much they inhibit the passage of current.
In the context of Ohm鈥檚 Law, resistance plays a key role in determining the relationship between voltage and current. Given the equation \( i = \frac{v}{R} \), you can see how resistance affects current flow:
  • As resistance increases, the current in a circuit decreases, when the voltage remains constant.
  • A reduction in resistance allows more current to flow if the voltage stays the same.
This inverse relationship makes resistance a crucial parameter for controlling current levels within electronic devices and circuits.

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