91Ó°ÊÓ

Faraday's Law of Induction is a fundamental principle that describes how electric currents can be induced by changing magnetic fields. In simple terms, when you have a loop of wire and the magnetic field inside the loop changes, an electric current will flow in the wire. This concept is extremely important in understanding how many types of electrical equipment operate, from transformers to electric generators.

For a rotating rod in a magnetic field, like the one in our problem, Faraday's law helps us calculate the induced voltage, or electromotive force (EMF), generated. In this case, the induced EMF is calculated using the formula: \( V = \frac{1}{2} B \omega L^2 \).

• \( V \): The electromotive force or potential difference
• \( B \): The magnetic field strength
• \( \omega \): The angular speed of the rod
• \( L \): The length of the rod

This equation shows that the induced voltage depends on three factors: the strength of the magnetic field, the speed at which the rod is rotating, and the square of the length of the rod. Understanding how these factors interact helps us determine how to achieve a particular voltage, which is crucial in real-world applications, such as generating electricity or in this exercise's context, achieving a potential difference sufficient to spark.

Angular speed is a measure of how fast an object rotates or spins. In physics, it's a key concept for systems involving rotational motion. The unit used for angular speed is radians per second (rad/s), which is different from linear speed measured in meters per second. In our exercise, angular speed is crucial as it directly impacts the potential difference created by the rotating rods.

To find the angular speed needed to produce a specific voltage, we use the modified Faraday's law formula: \( V = \frac{1}{2} B \omega L^2 \). Rearranging this to solve for angular speed \( \omega \), we have:
\[ \omega = \frac{2V}{B L^2} \] By substituting the known values for the potential difference (4500 V), magnetic field strength (4.7 T), and length of the rod (0.68 m), we can compute \( \omega \). Calculating each value carefully and ensuring units are consistent let us determine that the angular speed required is approximately 4160 rad/s.

Mastering angular speed calculations not only solve textbook problems but also plays an essential role in engineering and physics, allowing us to design components in engines, turbines, and other devices that rely on rotational motion.

Potential difference, often referred to as voltage, is a measure of the electrical potential energy between two points in a circuit. It's what "pushes" the electric charge to flow through a conductor, such as a wire or a rod. In this exercise, the required potential difference is what allows a spark to jump across the small gap between the ends of two rotating rods.

The potential difference needed to cause a spark depends on several factors, such as air resistance or the distance between charges. In air, it typically requires a minimum potential difference to initiate a spark. For our problem, the necessary potential difference is given as 4500 V, which is critical for the spark to occur.

By applying Faraday's Law, we calculated the potential difference generated by the rotating rods and set it equal to the required 4500 V to compute the correct angular speed for the spark. Understanding potential difference is vital in fields like electronics, as it enables safe and efficient design of circuits capable of handling specified amounts of power without causing unintended sparks or electric arcs.