Question

A uniform non-conducting ring of radius 2.68 cm and total charge 6.08 µC rotates with a constant angular speed of 4.21 rad/s around an axis perpendicular to the plane of the ring that passes through its center. What is the magnitude of the magnetic moment of the rotating ring?

Answer 1

The magnitude of the magnetic moment for a uniformly charged rotating ring is calculated using the current and the area, based on the ring's total charge, radius, and angular speed. By converting the given values to standard units and applying the formula, we can compute the magnetic moment.

Step-by-step explanation:

The magnitude of the magnetic moment (μ) of a rotating charged ring can be determined using the formula μ = I * A, where I is the current and A is the area of the loop. To find the current (I), we divide the total charge (Q) by the period of rotation (T), which is 2π over the angular speed (ω). Since the ring is uniform, the current is the same all around. The area (A) of the ring is π * r², where r is the radius. Therefore, the magnetic moment is given by μ = (Q / T) * (π * r²).

For the given values: charge Q = 6.08 µC, radius r = 2.68 cm, and angular speed ω = 4.21 rad/s, we first convert the charge to coulombs (1 µC = 1×10⁻¶ C) and the radius to meters (1 cm = 0.01 m). Then we calculate the magnetic moment.

Using the periodic relation T = 2π / ω, and substituting the values, the magnetic moment μ is computed as:

Answer 2

Answer: 1.72*10^-7

Step-by-step explanation:

Given

Radius of the ring, r = 2.68 cm = 0.0268 m

Charge on the ring, q = 6.08 µC

Angular speed of the ring, w = 4.21 rad/s

First, we know that the charge per unit area, σ = q / πr²

Also, the charge on ring of width, dr = σ⋅2πrdr

The Magnetic moment of this ring of width dr.dμ = i⋅A

If we integrate dr at R(top) and at 0(bottom), we get

∫dµ = ∫(R, 0) T⋅2πrdr.(w/2π).πr²

On finding at (R, 0), we get

μ = qwR² / 4

On substituting our values, we have

μ = (6.08*10^-6 * 4.21 * 0.0268) / 4

μ = (6.08*10^-6 * 0.113) / 4

μ = 6.87*10^-7 / 4

μ = 1.72*10^-7

The magnitude of the magnetic moment is 1.72*10^-7