For a solenoid... When the inner coil (with core) was used as the primary coil, was the ratio of the voltages equal to the ratio of the # of turns? explain any differences...

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Yes, the number of windings around the core is what determines the amount of current that is produced.  This is a direct proportionality from the primary coil to the secondary coil.  For example, if the primary coil has 24 loops around the core, and the secondary coil has 48 loops around the core, the induced electromagnetic force would be twice what the input current was on the primary coil.  This is called a "step-up" transformer, in that it induces twice as much current as was initially started with.  The ratio of loops is one to two (1:2), and that provides a current output that is twice the input current.  Of course, all you have to do is reverse the ratio, and you can easily produce a "step-down" transformer by having half the number of loops on the secondary coil as you have on the primary coil.  This would reverse the proportionality, giving a two to one ratio (2:1), cutting the current in half.

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