Question 2

2A) An electromagnetic plane wave, with wavelength $\lambda=2.00~{\rm m}$travels in free-space in the +x direction, with its electric vector ${\bf E}$, of amplitude $\vert{\bf E_0}\vert = 300~{\rm V/m}$ directed along the y-axis. .

i) [3 pts] What is the frequency of the wave?.

$$\nu~=~{c\over\lambda}~=~{2.998\times 10^8\over 2}~=~1.5\times 10^8~{\rm Hz}\ \ \ .$$

ii)[5 pts] What is the direction and amplitude of the magnetic field ${\rm B}$associated with this wave?.

$$\vert{\bf B}\vert~=~{ \vert{\bf E}\vert\over c}~=~{300\over 3\times 10^8}~=~10^{-6}~{\rm T}\ \ \ .$$

Pointing in the z-direction

iii) [6 pts] Find the Poynting vector of this wave.

$${\bf S}~=~{1\over\mu_0} {\bf E}\times {\bf B}~=~{300 10^{-6}\over 4\pi10^{-7}}{\bf\hat x}~=~{3000\over 4\pi}{\bf\hat x}~{\rm W/m^2}\ \ \ .$$

2B) [11 pts] An electromagnetic plane wave of average intensity Iis normally incident upon a vertical face of an indestructible, reflecting cube with sides of length 2a and uniform mass density $\rho$sitting on a no-slip surface. Find the minimum value of a for which the cube does NOT roll over due to the radiation pressure.

In order for the cube not to roll, the torque about the back edge in contact with the non-slip surface must vanish. At the point of rolling the torque due to the normal force from the surface vanishes, and what remains are the torques from gravity and from the laser. Firstly, from gravity, at the point of rolling we have

$$\vert\tau \vert~=~ M g a~=~8 \rho a^4 g\ \ \ ,$$

where M is the mass of the cube. Secondly, from the laser

$${\rm Power}~=~{\rm Intensity}\times {\rm Area}~=~c{d\vert{\bf p}\vert\over dt}\ \ \ ,$$

where the direction of the momentum transfer is in the direction of the energy absorption. For reflection, as we have here, there is an additional factor of 2 (this factor is modified when the cube begins to roll). Thus, we have

$$\vert\tau \vert~=~{\bf r}\times {\bf F}~=~a{d\vert{\bf p}\vert\over dt}~=~{8 I a^3\over c}\ \ \ ,$$

where we have used ${\rm Area}=(2a)^2$, is the area of a face of the cube. Thus for the cube not to roll we need

$$a\ge {I\over\rho g c}\ \ \ .$$