- #1
Adesh
- 735
- 191
Imagine that we have an electromagnetic wave or light propagating in x direction, and \mathbf{E} is oscillating in z direction and \mathbf{B} in y direction. The picture looks something like this
Now, if there exists a charged particle q on the xx axis at rest, then our B field can't do anything, but the electric field E will pull it upwards and as soon as it starts moving the magnetic field B comes into action and it goes like this $$ \mathbf{F_{mag}} = q (\mathbf{v} \times \mathbf{B}) $$ Since, E is going to cause a velocity in z direction i.e. perpendicular to magnetic field B , therefore $$ F_{mag} = q~v~B$$
$$ F_{mag} = q~v~\frac{E}{c}$$
$$F_{mag} = v~\frac{(qE)}{c}$$
$$ F_{mag} = \frac{F_{electric}~v}{c}$$
$$ F_{mag} = \frac{dW_{electric}/dt}{c}$$ and after doing this he writes
That light carries energy we already know. We now understand that it also carries momentum, and further, that the momentum carried is always 1/c times the energy
After considering this line, I thought the last equation could be written as $$ \frac{dp}{dt} = \frac{1}{c} ~ \frac{dW}{dt}$$
$$ dp = \frac{dW}{c} $$
$$ p= \frac{E}{c}$$
But I find this thing sloppy because when I wrote F_{mag} = \frac{dp}{dt} it was for the particle, that is force applied on the particle was equated with the rate of change of particle's momentum and not the light's momentum and similarly dW is the energy imparted to the particle and not the energy of light itself.
So, these are my problems. I earnestly request you to please help me over here.
Now, if there exists a charged particle q on the xx axis at rest, then our B field can't do anything, but the electric field E will pull it upwards and as soon as it starts moving the magnetic field B comes into action and it goes like this $$ \mathbf{F_{mag}} = q (\mathbf{v} \times \mathbf{B}) $$ Since, E is going to cause a velocity in z direction i.e. perpendicular to magnetic field B , therefore $$ F_{mag} = q~v~B$$
$$ F_{mag} = q~v~\frac{E}{c}$$
$$F_{mag} = v~\frac{(qE)}{c}$$
$$ F_{mag} = \frac{F_{electric}~v}{c}$$
$$ F_{mag} = \frac{dW_{electric}/dt}{c}$$ and after doing this he writes
That light carries energy we already know. We now understand that it also carries momentum, and further, that the momentum carried is always 1/c times the energy
After considering this line, I thought the last equation could be written as $$ \frac{dp}{dt} = \frac{1}{c} ~ \frac{dW}{dt}$$
$$ dp = \frac{dW}{c} $$
$$ p= \frac{E}{c}$$
But I find this thing sloppy because when I wrote F_{mag} = \frac{dp}{dt} it was for the particle, that is force applied on the particle was equated with the rate of change of particle's momentum and not the light's momentum and similarly dW is the energy imparted to the particle and not the energy of light itself.
So, these are my problems. I earnestly request you to please help me over here.
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