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Image we have an ultra-high intensity, ultra low frequency laser, with wattage on the order of terawatts and a wavelength on the order of a lightsecond. We rotate it that the electric field component is oriented on the $\hat z$ axis, then fire it at a macroscopic block with a positive electric charge. Because of the low frequency the block will experience an electric that doesn't immediately change direction, and because of the high intensity the field will be very strong. So from this naive understanding of classical physics, the block will briefly levitate.

Except this blatantly contradicts both QM (Compton scattering) and multiple macroscopic experiments (like solar sails), which both say that the block will be pushed in the direction of the laser. What assumptions in the original problem are missing/wrong?

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There will indeed be a small motion transverse to the beam. The motion will also result in a magnetic force along the beam. Now, imagine half a wave cycle later, the electric (and magnetic) field has reversed direction, the transverse motion is reversed, however the magnetic force is not reversed, it is still directed along the beam, because both the (transverse) velocity and the magnetic field have flipped sign. Over many cycles, the transverse motions cancel each other out, while the longitudinal motion does not.

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Block doesn't levitate because the total force due to the electric field is zero as block has equal amount of positive and negative charges.According to classical physics laws if block is non-conducting, it wont even move in the direction of laser. But if its conducting, electric field will give rise to a current in its face on which laser is incident.As electrons start moving in direction opposite to electric field, they will experience magnetic force F=-e(v*B) which is in the direction of propagation of laser as you can verify using right hand rule and due to this force things move in the direction of incident light. and for the same reason when charges start moving in transverse direction due to electric field of light,,,they experience magnetic force due to their velocity which obviously is not transverse to the light direction,

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Part of the premise is we "fire it at a macroscopic block with a positive electric charge.", so the forces from positive and negative charge shouldn't balance out. –  Hovercouch Sep 26 '13 at 15:44
in that case, if wavelength is the order of lightseconds,, then frequency will be the order of 1Hz,, and in steady state, block will move almost in a cycloid shape curve,, going up and down and also drifting in the the direction of laser,, but suerly it will not just levitate at one point,, –  Salman Azmat Sep 26 '13 at 16:41
I get that. My problem is that this contradicts both QM and experiment, which say the block will move in the direction of the beam. –  Hovercouch Sep 26 '13 at 16:44
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