Consider relative motion between two reference frames.
The velocity of point C in reference Frame A is the velocity of C in Frame B
plus the relative velocity between the frames.
We can take the time derivative of both sides of this equation.
If the relative velocity between the frames is constant, the acceleration of C as seen by both frames is the same.
This means that even though the relative frames do not measure the same speed for a third moving frame,
they must agree on the magnitude of force felt on a particle in a third frame.
Consider the case where a charged particle is moving in a magnetic field created in a lab.
In the rest frame of the lab, the charged particle moving through the field feels a Lorentz force
which deflects the particle. In a frame comoving with the particle,
the charged particle is at rest and therefore does not feel a Lorentz force.
Observers in both frames must agree regarding the force on the particle and the
deflection of the particle.
The diagram on the left illustrates a charged particle moving with velocity v in a magnetic field created in Frame A.
Observer A measures a Lorentz force FA acting on the particle.
The diagram on the right depicts Frame B, moving relative to Frame A,
with the same velocity as the charged particle.
Observer B must measure the same force on the charged particle but it cannot be a Lorentz force,
since the particle is at rest in that frame.
We conclude that the force accelerating the particle must arise from an electric field.
The motion of the reference frame relative to the source of the field determines whether the
field detected is an electric field or a magnetic field.
In general, an experimenter in Frame A can create an electric field and a magnetic field.
The force felt by a moving charged particle in Frame A will be the vector sum of the Lorentz force
and the Electric Force. Observer B in Frame B comoving with the particle will detect a force with the
same magnitude and direction,
but attribute it only to an electric field.
Because the forces felt in the reference frames must be equal, we can write the electric field in Frame B
as the vector sum of the electric Field in Frame A plus the Lorentz force per charge in Frame A.
This equation gives the transformation of the electric and magnetic fields in Frame A into the electric
field in Frame B.
Now consider a charged particle at rest in Frame A. The charge particle creates an electric field.
Since it is not moving in Frame A, it does not create a B field in Frame A.
The electric force does not depend on velocity
so the form of the electric force equation is the same for the two frames.
However, Observer B does detect a B field arising, since in that frame the particle appears to be
moving backwards at the relative velocity. We find the B field from the moving charged particle
in Frame B by using the Biot-Savart law.
With some algebraic manipulation, we can see the expression for the magnetic field seen by Observer B
can be written in terms of the electric field of Frame A.
The Biot-Savart Law for a moving charged particle is equal to the Coulomb electric field of a
stationary point transformed into a reference frame moving at constant speed.
For the general case where Frame A has created both an electric field and a magnetic field,
the full transformation takes this form.
We summarize our results in the equations above, known as the Galilean field transform equations,
where the fields are measured at the same point by frames moving at relative velocities.
These equations are valid where vAB << c and verify that the electric field and
the magnetic field are two aspects of a single field: the electromagnetic field.
Practice problems
1. The image depicts Frame A in a lab with electric and magnetic fields as shown.
Frame B that measures a magnetic field that has greater magnitude than the magnetic field measured
in the lab frame.
In which of these directions could Frame B be moving?
A. Positive x-direction
B. Negative x-direction
C. Positive y-direction
D. Negative y-direction
2. A uniform 1.2 MV/m electric field in the positive z direction is generated in the region surrounding a
space station. What are the E and B fields measured by a passing rocket traveling at 2500 km/s
in the positive x direction?
3. In a lab frame, parallel E and B fields have been generated in the positive x direction. A proton
is moving in the fields with a velocity v in the positive y direction.
Consider the case where E = 12 kV/m, B = 0.14 T and v = 1.8 x 105m/s.
A. What is the magnitude of the E field in the proton's frame?
B. What is the angle of the E field with respect to the positive x-axis?
Take counterclockwise to be positive.
