Very rapidly switch polarity of an electromagnet

So I am working on a project of basically turning a 6.5 hp mini bike engine into an electromagnetic piston engine by putting an electromagnet on the top of the cylinder and a permanent magnet on the piston. My problem here is that I am trying to replicate the speeds of which an engine would normally operate, in this case it is a max of 4000 rpm. Is it possible to use the PWM within arduino and a MOSFET to achieve these speeds? What are some other options in order to achieve what I am looiking for?

Not likely. Magnetism needs quite a short distance to work. The distance would be counted in millimeters for "normal" electromagnets. Using exceptionally strong NEUDYM magnets You can talk about centimeters. However such magnets are large.

Your problem is much larger than You tell in the post.....
Why would PWM help? Pulsing synchronised with the actual RPM might work in theory... That calls for a variable frequency, not a variable duty cycle.

I am very new to arduino so I am not sure if PWM would help at all, Just from what I have been reading on motors it would make since to do something similar since they are essesntially electromagnets. How can I utilize pusling synchornised?

You would do best by learning about magnetism and electromagnetism. Your idea, as I get it from Your post, is ways from possible.

6.5 hp is about 5kW so you may have some difficulty matching that power with the experiment you have described.
Look at H-bridge circuits for rapid polarity reversal. In electronics terms, 5000RPM is slow.

Maybe look at something smaller ro start with https://nl.pinterest.com/pin/181129216250995004/

Not convinced that's going to fly - big airgap in the magnetic circuit will mean extremely low efficiency. Real solenoids work by having variable reluctance along the stroke and as small an air-gap as possible. The lower the average reluctance the larger the forces possible.

Re-engineering the cylinder as a linear (voice-coil) motor might be workable, but its a complete reworking of the engine.

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