Why is magnetic field not accounted for in the theory of CRO

Question

I’m reading the chapter on Oscilloscopes in Cooper’s Electronic Instrumentation and Measurement Techniques.

Nowhere in the theory of the working of CRO (like deflection sensitivity) does he take into account the effect of magnetic field that should definitely be present due to rapidly changing very high voltages (~1000 V) applied on the vertical plates (~10 MHz or more).

But I’m aware that the speed of electrons in CRO is huge (~1/10th the speed of light). So even small magnetic fields should affect the motion greatly. (Magnetic force has a cross product of velocity and field.)

So why is neglecting it reasonable?

Answer 1

the speed of electrons in CRO is huge (~1/10th the speed of light). So even small magnetic fields should affect the motion greatly.

Magnetic force is proportional to velocity, but the force required to produce the same deflection radius is proportional to velocity squared, so a faster beam will be deflected less than a slower beam.

So why is neglecting it reasonable?

The force on the electron beam is perpendicular to the magnetic field, so to deflect the beam across the tube face the magnetic field has to be parallel to it. However when AC current is passed through a capacitor it generates a circular magnetic field between the plates. As the beam passes between the plates it is deflected first one way and then the other as it crosses magnetic fields going in either direction, so overall the deflections tend to cancel out.

In a typical Oscilloscope tube the capacitance between deflection plates is only 1~2pF, so the displacement current at normal operating frequencies is small. The magnetic field produced is equivalent to the field around a single wire passing the same current, so if the plates were eg. 10mm wide and had a capacitance 2pF, and 100V was applied at 1GHz causing a current of ~1A, then ~0.2 Gauss would be produced - less than half that of the Earth's magnetic field. And much of that would cancel out.

At extremely high frequencies the magnetic deflection could be large to have a noticeable effect, but then other effects of the high frequency also become apparent. At GHz frequencies the electrostatic deflection amplitude reduces because the electrostatic field changes as the electrons are passing between the plates. At even higher frequencies deflection varies depending on the phase when the beam exits the plates.

At frequencies high enough for the magnetic field to become noticeable the tube has probably already lost its ability to faithfully reproduce any waveform other than a sine wave, and the amplitude is highly frequency dependent even without considering magnetic effects.

Answer 2

So why is neglecting it reasonable?

For the earths field

\$ \vec{F}= q(\vec{E} + \vec{v}\times\vec{B})\$

A 1000V field compared to 25uT (earths field) crossed with 3e7m/s isn't negligible (750 from VxB when compared with 1000 from E). But the electron is being accelerated almost from rest so the E term is constant over the trajectory of the electron, whereas the VxB term increases as the velocity increases.

Most of the time, the magnetic field would not be aligned perpendicular to the magnetic field. Many CRT's (or CRO's) have magnetic shielding, and CRT's are calibrated which can reduce the effects from local static magnetic fields.

If there is a small field, and the CRT is shielded, then I think the contribution from the magnetic field would be neglectable.

Anything stronger than the earths magnetic field and it does affect the path of the electrons

Source: https://www2.physics.ox.ac.uk/accelerate/resources/demonstrations/cathode-ray-tube