You're an electrical engineer designing an alternator (the generator that charges a car's battery). Mechanical engineers specify a 10-cm-diameter rotating coil, and you determine that you can fit 250 turns in this coil. To charge a 12 -V battery, you need a peak output of \(14 \mathrm{V}\) when the alternator is rotating at \(1200 \mathrm{rpm}\). What do you specify for the alternator's magnetic field?

Electromagnetic induction is the process of generating electric current from a changing magnetic field. In the context of an alternator—which is essentially a generator used in cars to charge the battery—electromagnetic induction is at the heart of its operation.

When a conductor, such as a loop of wire, is moved within a magnetic field, a difference in electrical potential, or voltage, is induced across the conductor. This principle was discovered by Michael Faraday, and is mathematically described by Faraday's law of induction. The law states that the induced voltage in any circuit is proportional to the rate of change of the magnetic flux through the circuit. In simple terms, faster changes in the magnetic field or a circuit with more loops will result in a higher voltage.

For the design of an alternator, understanding electromagnetic induction is crucial in determining how much voltage can be generated and how to maximize this output to meet the voltage requirements of the car's battery. By calculating the interactions between the coil of wire, the speed of rotation, and the strength of the magnetic field, electrical engineers like you can optimize the alternator design.

The design of an alternator is centered around maximizing the efficiency of electromagnetic induction. Let's break down the elements from the exercise that influence alternator design.

• Coil Diameter and Turn Count: The diameter of the coil and the number of turns of wire within the coil are key aspects of the design. More turns can generate a greater voltage for a given magnetic field strength.
• Rotation Speed: The rotation speed of the coil—measured in rounds per minute (rpm)—directly impacts the frequency of the alternating current produced. Higher speeds can produce more voltage but also lead to more mechanical wear.
• Magnetic Field: The strength of the magnetic field within which the coil rotates is a variable that needs to be carefully controlled to meet the voltage output requirements of the alternator.

The battery in a car requires a specific charging voltage; hence the alternator needs to produce a slightly higher voltage to ensure efficient charging. As specified in the exercise, you need a peak output of 14V to charge a 12V battery at 1200 rpm. Taking into account the coil size and the number of turns, you can adjust the strength of the magnetic field to achieve the necessary voltage, as demonstrated by the textbook solution.

Calculating the magnetic field strength is a critical step in alternator design and the application of Faraday's law of electromagnetic induction.

From the exercise, we were provided with the requirement to generate a peak voltage of 14V at a rotation speed of 1200 rpm for a coil with a 10-cm diameter and 250 turns. To convert rpm to Hz, which is necessary for using Faraday’s law, you divide by 60 (since there are 60 seconds in a minute) to get the frequency of rotation.

After establishing the coil's area and having the frequency, you would apply these values to Faraday’s law formula: \( V = 2 \pi f N B A \). This equation shows that the induced voltage (\(V\)) is equal to the product of the angular frequency (\(2 \pi f\)), the number of turns (\(N\)), the magnetic field strength (\(B\)), and the area of the coil (\(A\)). Solving for \(B\) using the given values, as detailed in the solution, gives us the exact magnetic field strength required to achieve the desired voltage output.

Understanding the calculations behind the magnetic field is not only instrumental for the alternator design but also allows for the prediction and adjustment of the alternator’s performance based on physical variables such as the number of turns in the coil or the coil's size. The ability to manipulate these variables and calculate their effect on the alternator's output enables engineers to tailor the design to specific requirements.