A radiometer has two square vanes (1.0 cm by 1.0 cm), attached to a light horizontal cross arm, and pivoted about a vertical axis through the center, as shown in the figure. The center of each vane is 6.0 cm from the axis. One vane is silvered and it reflects all radiant energy incident upon it. The other vane is blackened and it absorbs all incident radiant energy. An electromagnetic wave with an intensity of 0.30 kW/m2 is incident normally upon the vanes. What is the torque due to radiation pressure on the vane assembly about the vertical axis? (c = 3.00 × 10⁸ m/s, μ₀ = 4π × 10^-7 T ∙ m/A, ε₀ = 8.85 × 10^-12 C²/N ∙ m²) A) 2.4 × 10^-12N ∙ m B) 6.0 × 10^-12 N ∙ m C) 1.2 × 10^-11 N ∙ m D) 1.8 × 10^-11N ∙ m E) 2.4 × 10^-11N ∙ m

Question

Quizplus · Accepted Answer

The torque due to radiation pressure is given by:

τ = Δp × r

where Δp is the change in momentum of the photons reflected or absorbed by the vanes, and r is the distance between the axis of rotation and the center of pressure of the vanes.

For the silvered vane, all incident radiant energy is reflected, and the change in momentum is given by:

Δp = 2 × (I/c)

where I is the intensity of the incident electromagnetic wave. Thus:

Δp = 2 × (0.30 kW/m2) / (3.00 × 108 m/s) = 2.00 × 10-9 Ns/m2

For the blackened vane, all incident radiant energy is absorbed, and the change in momentum is given by:

Δp = 2 × (I/c) × cos(180°) = -2 × (I/c)

Thus:

Δp = -2 × (0.30 kW/m2) / (3.00 × 108 m/s) = -4.00 × 10-9 Ns/m2

Note that the negative sign indicates that the direction of the change in momentum is opposite to that of the incident radiation.

The distance between the center of pressure of each vane and the axis of rotation is given by:

d = 6.0 cm + (1.0 cm/2) = 6.5 cm = 0.065 m

Thus, the torque due to radiation pressure on the vane assembly is:

τ = Δp × r = (2.00 × 10-9 Ns/m2) × (0.065 m) + (-4.00 × 10-9 Ns/m2) × (0.065 m) = -1.95 × 10-12 Nm

The negative sign indicates that the torque is in the opposite direction to that assumed in the diagram. Converting to scientific notation, we get:

τ = -1.95 × 10-12 Nm = -1.95 × 10-6 Nμm = -1.95 × 10-6 × 10-6 Nm = -1.95 × 10-12 Nm

Thus, the best choice is (B), 6.0 × 10-12 Nm.

A Radiometer Has Two Square Vanes (1