Can a shorter magnet produce the same current as a taller magnet in a solenoid?

[jfowle]

Greetings,

The other day while working on my car I began thinking about solenoids and how they work, but I think I may have confused myself. It's been a while since I've studied magnetism, but I remember that the current through a wire loop, and thus a solenoid as well, is dependent on the change in magnetic flux through the cross-sectional area of the loop.

So, if you have a cylindrical magnet in the solenoid that is the same height as the solenoid and you pull it out, you should get a certain current through the solenoid wires. If instead you have a cylindrical magnet with the the same magnetic flux density as the first magnet, but much shorter than the height of the solenoid (so, basically a disk magnet), would it produce the same current if it traveled the length of the solenoid?

I thank the physics gods and all who give any thought to the questions that come from my feeble mind.

 

[kuruman]

So, if you have a cylindrical magnet in the solenoid that is the same height as the solenoid and you pull it out, you should get a certain current through the solenoid wires. If instead you have a cylindrical magnet with the the same magnetic flux density as the first magnet, but much shorter than the height of the solenoid (so, basically a disk magnet), would it produce the same current if it traveled the length of the solenoid?

The same current is a loaded expression because the current in each case will be time-dependent. However, you will get the same charge to flow through the solenoid in each case. If you arrange things so that the same charge flows over the same time interval, then you can argue that the same average current flows in the solenoid. The reason you have the same amount of charge is this.

The current in the solenoid at any time is given by $$I=\frac{1}{R}\frac{d\Phi_M}{dt}$$where $R$ is the resistance of the solenoid and $\Phi_M$ the magnetic flux through it. The amount of charge that flows in time interval $dt$ is$$dq=Idt=\frac{1}{R}\frac{d\Phi_M}{dt}dt$$Thus, the total charge that flows through the solenoid is obtained by integrating$$\int dq=\frac{1}{R}\int\frac{d\Phi_M}{dt}dt$$ $$\Delta q=\frac{1}{R}\Delta \Phi_M$$So if you bring in each magnet from very far away, push it through the solenoid and back out to very far away, the total charge that will flow through the solenoid will be the same in each case as long as the total magnetic flux change is the same.

I don't know any physics gods, but the spirit of Michael Faraday is smiling benignly upon you.