Can I PWM this electromagnet at audio frequencies?
If you want to make the field change quickly and if the inductance is relatively large, the only way to make that happen is with a lot of volts. It sure ain't 1uH. It could easily be ~1H. 

You can experimentally try it (with a flyback diode and a low duty cycle pulse) to see how rapidly the current builds up (on an oscilloscope). If it is too slow, you might have to do something like use a higher voltage supply with a resistor (dumping copious heat) in series. If you're just doing a test that might be fine. The field strength is proportional to the current through the coil. Similarly, tying the flyback diode to a high voltage supply will allow the field to collapse relatively fast. 

I'm not sure what you are trying to accomplish with an H-bridge unless you are planning to put a permanent magnet on the bar. For a ferroelectric bar the electromagnet "sucks" with either polarity of current.