How can you verify that a current carrying wire produces a magnetic field with the help of an experiment?

Hint: Recall Faraday's experiment, he accidentally noticed some sort of deflection in the magnetic needle when the needle of the compass was close to a wire carrying current. This amused Faraday for a moment then he started doing the experimenting by moving a compass near the current carrying wire.

Complete step by step answer:
Here two methods are described, both are effective in their own way.
Method 1:
-First, we take cardboard and make a hole through it.
-Pass a copper wire through the hole.
-Connect the two ends of the wire with the battery through a switch.
-Place a compass around the hole of the cardboard.
-Switch on the current, when the current flows the magnetic needle deflects.
-We can verify this by changing the direction of current. As the direction of the flow of current changes the direction of the compass needle reverses.
-So, we can conclude the magnetic field is produced around a current carrying conductor.

In the above diagram we can see that the compass gets aligned in the direction of the magnetic field thus from this method we can also determine the direction of the magnetic field.

Method 2:
-First, we take cardboard and make a hole through it.
-Pass a copper wire through the hole.
-Connect the two ends of the wire with the battery through a switch.
-Sprinkle some iron filings in the cardboard around the wire.
-Switch on the current, when the current flows the iron filings align themselves as in a pattern similar to that of a magnetic field line.
-So, we can conclude the magnetic field is produced around a current carrying conductor.

We can see from the above diagram that the major fillings align in the lines of the magnetic field whereas the rest don't; this indicates a sign of presence of impurities.

Note: Both methods are effective in determining the presence of a magnetic field due to a current carrying wire, but the second one can’t tell us the direction of the magnetic field. Even though the second method is not useful in determining the direction of magnetic fields, this method can be very useful to separate very minute non magnetic particles from magnetic particles effectively by passing a high value of current.