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A rotating charge refers to a situation where a charged particle is moving along a circular path. In physics, when such motion occurs, it can produce a magnetic field similar to a loop of current. This is because, as the charge moves in a circle, it effectively acts like a loop of electric current. Imagine tying a small charge to the edge of a spinning wheel. If the wheel spins steadily, the charge rotates around a fixed center point. This movement is essential for calculating properties like the magnetic moment.
The exercise involves a charge of 4.0 × 10^-6 C on a conducting sphere, which is attached to a rod, rotating around an axis at the speed of 150 rad/s. Here, the path traced by the charge is a circle with a radius equal to the length of the rod, 0.20 m. These elements combine to form a current-like behavior in terms of physics, leading us to the concept of magnetic moment calculation.

Angular speed (\( \omega \)) characterizes how fast an object rotates or revolves around a central point. It is typically measured in radians per second (\( \ ext{rad/s} \)). For a body rotating in a circle, like the charge in the problem, angular speed defines how quickly it moves along its circular path.
This concept plays a crucial role in understanding the behavior of the rotating charge. Given \( \omega = 150 \ ext{ rad/s} \), this tells us how many radians the charge covers each second while performing its circular movement.

• If you think of the circle as a clock, a complete rotation would be \( 2\pi \) radians (since a circle has \( 360^{\circ} \), and in radians this is \( 2\pi \)).
• Given this angular speed, you can calculate the time it takes for one full revolution by determining the period (T).

This relationship between angular speed, period, and radius is vital for computing other related phenomena, such as linear speed and magnetic moments.

In physics, a current loop is a concept where a steady flow of electric charge circulates in a closed path or loop. This creates a magnetic field through a principle known as Ampere's Law. Here, the rotating charge effectively simulates such a current loop.
To visualize, imagine the charged sphere at the tip of our 0.20 m rod. As it spins, the motion is akin to the charges moving through a circular wire. The speed and frequency of these rotations develop an equivalent current, even though there is no actual wire.

• The equivalent current (\( I \)) is calculated using the formula \( I = \frac{q}{T} \), where \( q \) is the charge and \( T \) is the period.
• The charge completes one rotation per period, creating a complete current loop effect.

This creates the foundation for further calculations of the magnetic moment, as this imaginary 'loop' mimics the effect of real current loops in magnetic field generation.

The magnetic moment is a measure of the strength of a magnetic source. For our rotating charge, the magnetic moment (\( \mu \)) can be computed by considering the circle as both an area and a current source. The key formula here is \( \mu = I \times A \), where \( I \) is the effective current and \( A \) is the area of the circle.
The area of the circle is calculated as \( A = \pi r^{2} \). Using the formula, substitute \( r = 0.20 \ ext{ m} \):

• Find area \( A = \pi \times 0.20^2 = 0.04 \pi \ ext{ m}^{2} \).
• The effective current \( I \) comes from charge divided by the computed period \( T \).
• \( \mu \) is found by substituting \( I \) and \( A \) into the magnetic moment formula.

Thus, by combining these calculations, we determine the magnetic moment of the rotating charge in the exercise as \( \mu = 1.2 \times 10^{-5} \ ext{ A} \cdot \ ext{m}^{2} \). This represents how effectively the rotating system simulates a magnetic field.