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A solenoid is connected to a battery so ...

A solenoid is connected to a battery so that a steady current flows through it. If an iron core is inserted into the solenoid, then

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The current will decrease. This is because on inserting an iron in the solenoid, magnetic field and hence magnetic flux linked with the solenoid increase. As per Lenz's law, the emf induced in the solenoid will oppose this increase, which can be achieved by a decrease in current.
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