You can use small coil or 1 ring and try to see indused V. If You have magnet with iron core   it's immpossible to get 1 ms pulse of field. A Hall probe can work with AC field up to 1-10 mks

Most of Hall sensors have typically a response time better than 10 microseconds with a correct load and amplifier circuitry

If you use a rather big electromagnet with a ferrite core it could be impossible to pulse correctly the field in your 5 mS with the waited shape because the L.di/dt reaction effect.

I'm sure that faster hall sensors are available. 5 ms is nothing to be considered 'fast'. But you might run into signal processing issues as the signals of the faster sensors tend to be smaller :)

But: IMHO it is not necessary to measure the pulsed field directly! as it is proportional to the excitation current, it should be possible to calibrate your circuit with different levels of DC current resulting in a calibration curve B = f(I). Then you 'only' have to measure the current, not the field itself.

This should work, EXCEPT when you have "field absorbers" that are selectively weakening field changes (eg. eddy currents in a metallic plane). In this case I'd suggest an additional AC calibration with different levels and frequencies of AC (pure sine) current - measuring the field with a probing coil. Be careful not to load the source too much with the probing coil - it is also a "field absorber".

Be precise and have fun!

I would start by talking to a researcher who is knowledgeable regarding low thrust electric propulsion systems.  Perhaps Prof. Lyon King at Michigan Tech Univ in Houghton, MI (the UP).

I would also recommend measuring current to determine your field, after first calibrating the field per unit current at DC. We use a Rogowski coil (made by PEM corporation) in our lab to measure AC current. They have rapid response time that should be suitable for measuring pulses as well.

You can use magnet optical effect to do the calibration.  Just take one material having Faraday rotation(response) and calibrate its response to DC field. Then use it to calibrate your pulse field.   

One could suspend a small (2-3cm) piece of high frequency ferrite (a toroid core for example) in front of the magnet. Apply dc excitation, measure deflection of ferrite. Measure static field. Calculate force on "pendulum" per Gauss. Apply pulse of known width. Measure initial deflection of pendulum (treat it as a ballistic pendulum). Calculate b-feld. That's how we did it 100 years ago. Note that perm and susceptibility of the pendulum wash out of the equations -- especially if you use a ferrite with a fairly linear BH curve and calibrate (DC) in the same ballpark as the pulsed B.

You can make a Bdot probe (google it) and use to measure the solenoid field. Depending upon your signal size and how many loops you make the Bdot with you can just put it straight into a scope and integrate it numerically. An easy way to calibrate it would be to turn on your solenoid with a low level current step (say 1 amp driven from a current limited supply) with a similar risetime to your actual signal and when integrated and multiplied by the calibration factor it should give you the field measured by your hall probe. If calibrated properly the Bdot can be used to measure extremely fast magnetic fields.

Thank you everyone for your answers. It helped a lot. I calibrated the field by two different methods. The calibration by using a detection coil of smaller size than the magnet worked fine and it was possible to calibrate the field. Even it could be checked down to 500 microsec pulse width.