PHY2061 Enriched Physics 2 Lecture NotesRelativity 6
Relativity 6
Disclaimer:These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor.
Transformation of the Electric Field
Electric charge is invariant under motion. In other words, electric charge, like mass, is a property of a particle and is invariant under transformations. It is a Lorentz Invariant quantity. If this were not so, physics would look different in different reference frames. The light emitted by hydrogen would have different colors beyond just the usual Doppler shift. In fact, hydrogen might not even bind if the positive and electric charges did not transform the same way.
Thus, Gauss’ Law holds in all reference frames:
Two reference frames with a relative velocity between them would agree on the charge enclosed by a surface, even if the surface integral is computed using the coordinates of S or S’.
Consider 2 parallel sheets of charge infinite in extent with opposite charge density: . By Gauss’ Law in the rest frame of the two sheets, we can solve for the electric field arising from each sheet. Take as a Gaussian surface a box enclosing just one sheet or the other, with area on the top and bottom faces parallel to the sheet:
You can neglect the field arising from one sheet affecting the flux through the box enclosing the other sheet because the net flux will be zero through the surface (no charge enclosed from the other sheet). By the superposition principle, the two fields add, and we see that outside the two sheets, E = 0. Between the 2 sheets,:
Now consider a Lorentz Transformation to a frame where the 2 sheets are moving at a velocity v with respect to an observer. How do the previously considered Gaussian surfaces change? First of all, if we do not change the definition of the surface, the charge enclosed must remain invariant as stated before. However, the geometry of the box has changed. The dimension parallel to the direction of motion has contracted, so the area of the top and bottom surfaces has contracted.
In the S’ frame where the sheets are in motion, Gauss’ Law stated in that frame becomes:
But since the dimension parallel to the motion is length contracted, the area in the S’ frame is
So
In other words, the charge density has increased:
And the electric field perpendicular to the plates (and to the direction of motion) has increased:
If we had instead considered a Lorentz Transformation in a direction perpendicular to the plates (i.e. parallel to the electric field direction), there would have been no length contraction of the plate dimensions since they are perpendicular to the direction of motion. So the charge density would not change in the S’ frame, nor would the electric field:
Derivation of the Magnetic Force
Now consider two lines of charge (1-dimensional) moving in opposite directions with opposite charge density. In other words, a wire carries a current of positive charges to the right and negative charges to the left. Call this the lab frame S, and denote it with unprimed coordinates. In this frame, the velocities of the charges are equal and opposite:
The charge densities are also taken to be equal and opposite:
The net effect is a current moving to the right (positive x-axis direction):
The charge densities are measured in the lab frame. In the rest frame of the charges, the charge density would be different because of length contraction. We know from the previous section that for directions parallel to the direction of motion
So,
Now consider placing a test charge, q, next to these two lines of charge. In the lab frame, the charge densities are equal and opposite; so if the test charge is at rest in the lab frame, no electric force is felt since there is no net charge on the wire.
But consider the test charge moving with a velocity v parallel to the line charges in the lab frame (in the same direction as the positive charges). Let’s calculate the electric field from the wire acting on the charge in its rest frame. Thus, we want to consider a Lorentz Transformation to the rest frame of the particle.
To find the velocity of the line charges in this new frame where the test charge is at rest (call this frame S’), we must use the relativistic addition of velocity formulae:
The linear charge densities in this frame are increased relative to the rest frame of the moving charges by -factors depending on the velocity of the charges in the S’ frame:
Recall that is the charge density measured in the lab frame S, whereas is the charge density in the S’ frame moving with the test charge.
Now since , the charge densities will not be equal and opposite (because of differing amounts of length contraction of the original charge densities). Thus, there now will be a net charge on the wire, and an electric force on the particle!
The net charge density can be calculated:
which is indeed not zero.
We can use this net charge density to calculate the electric field at the position of the test charge (which is at rest in this frame):
Appling Gauss’ Law to an infinitely long line charge, we consider a cylindrical surface:
The force on a test charge of charge q is toward the wire (since the wire has a net negative charge in this frame) if q>0:
Thus, the particle which was originally at rest, will accelerate toward the wire (in the direction perpendicular to the Lorentz Transformation).
Now, let’s transform this force back to the original frame where the test charge was in motion and the wire had no net charge density. In this frame, the force perpendicular to the direction of the Lorentz Transformation direction is:
, so
Now is just the current carried in the wire. Also, note that . So,
where we have defined a new constant:
The term in parentheses can be defined to be the “magnetic field” arising from a current in a wire:
We have just derived the force acting on a charged particle when it moves at a velocity v with respect to a current i. Note that the force is in a direction perpendicular to the velocity direction. This force is called the magnetic force, with an associated magnetic field. However, it really is nothing new. In the rest frame of the test charge, only an electric field existed. But when transformed to the lab frame, the electric field changed into a magnetic field that only operates on charges with non-zero velocities.
The magnetic force equation can be written more generally in vector form as:
The force is the cross-product of the velocity and magnetic field vectors, multiplied by the charge of the particle:
When combined with the electric force law, we have the full Lorentz Force equation:
D. AcostaPage 111/13/2018