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Chapter 2: Problem 12

Ohm's law says that when electric current is flowing across a resistor, then the voltage \(v\), measured in volts, is the product of the current \(i\), measured in amperes, and the resistance \(R\), measured in ohms. That is, \(v=i R\). a. What is the voltage if the current is 20 amperes and the resistance is 15 ohms? b. Find a formula expressing resistance as a function of current and voltage. Use your function to find the resistance if the current is 15 amperes and the voltage is 12 volts. c. Find a formula expressing current as a function of voltage and resistance. Use your function to find the current if the voltage is 6 volts and the resistance is 8 ohms.

Short Answer

Expert verified
a. 300 volts; b. 0.8 ohms; c. 0.75 amperes.

Step by step solution

01

Understanding Ohm's Law

Ohm's Law is given by the formula \( v = iR \), where \( v \) is the voltage, \( i \) is the current, and \( R \) is the resistance.
02

Calculating Voltage

To find the voltage when the current is 20 amperes and the resistance is 15 ohms, substitute these values into the equation \( v = iR \): \[ v = 20 \times 15 = 300 \] So, the voltage is 300 volts.
03

Deriving the Resistance Formula

To express resistance as a function of current and voltage, rearrange the formula \( v = iR \) to \[ R = \frac{v}{i} \] This shows that resistance \( R \) can be found by dividing voltage \( v \) by current \( i \).
04

Calculating Resistance

Substitute the given values of voltage (12 volts) and current (15 amperes) into the resistance formula \( R = \frac{v}{i} \): \[ R = \frac{12}{15} = 0.8 \] Thus, the resistance is 0.8 ohms.
05

Deriving the Current Formula

To express current as a function of voltage and resistance, rearrange the formula \( v = iR \) to:\[ i = \frac{v}{R} \] This shows that current \( i \) can be determined by dividing voltage \( v \) by resistance \( R \).
06

Calculating Current

Substitute the given values of voltage (6 volts) and resistance (8 ohms) into the current formula \( i = \frac{v}{R} \): \[ i = \frac{6}{8} = 0.75 \] Therefore, the current is 0.75 amperes.

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Key Concepts

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

Voltage
Voltage is the potential difference between two points in an electrical circuit. It's like the pressure that pushes electric charges through a conductor. Voltage is measured in volts, and it is represented by the letter \( v \). Think of voltage as the force in a water hose that pushes water along. The higher the voltage, the greater the force that pushes the current through the circuit.
  • Voltage is crucial for the function of all electronic devices, as it determines how much energy can be transferred to different parts of a circuit.
  • When you plug a device into an outlet, you are tapping into the voltage provided by the electrical grid, which enables it to function.
In practical terms, increasing the voltage in a circuit means more energy is available to move electric charges through the circuit's resistance. For example, if the current is 20 amperes and the resistance is 15 ohms, using Ohm's law \(v = iR\), the voltage would be \(v = 20 \times 15 = 300 \) volts.
Current
Current refers to the flow of electric charge in a circuit. It's similar to water flowing through a pipe. Current is measured in amperes or amps, denoted by the letter \( i \). The more electric charges that pass a point in the circuit per second, the higher the current.
  • Current is what powers devices and makes them work. It's the actual movement of electrons through a conductor.
  • Making a connection between electrical wires or turning on a device allows current to flow.
Consistent current flow is necessary for electronics to function properly, similar to how water needs to flow through a pipe to reach a destination. If you rearrange Ohm鈥檚 Law to solve for current, you get \(i = \frac{v}{R}\). For instance, if the voltage is 6 volts and the resistance is 8 ohms, the current would be \(i = \frac{6}{8} = 0.75\) amperes.
Resistance
Resistance limits the flow of electric current in a circuit. This is akin to the friction that water experiences when flowing through a pipe. Resistance is measured in ohms, represented by the letter \( R \). Higher resistance means less electric current can pass through a conductor.
  • Components like resistors are used in circuits to control current flow, ensuring that devices don't receive too much current and become damaged.
  • The body's resistance plays a crucial role in controlling the electric current flowing through when in contact with a power source.
Using Ohm鈥檚 law, resistance can be calculated with the formula \(R = \frac{v}{i}\). For example, with a voltage of 12 volts and a current of 15 amperes, the resistance would be \(R = \frac{12}{15} = 0.8\) ohms. Understanding resistance helps us design circuits that function safely and efficiently.

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