Introduction:
A vibration magnetometer is a device used to measure the strength and direction of a magnetic field. It consists of a small bar magnet suspended by a thin fiber or spring, allowing it to oscillate freely. The time period of these oscillations can be used to determine the strength of the magnetic field.

Given Information:
- Time period of the magnet's oscillations in Earth's magnetic field: 2 seconds
- Strength of Earth's horizontal magnetic field: 24 microtesla
- Strength of the horizontal magnetic field produced by a current carrying wire in the opposite direction: 18 microtesla

Analysis:
When the magnet is placed in Earth's magnetic field, it aligns itself with the field lines and executes oscillations. The time period of these oscillations is directly proportional to the strength of the magnetic field.

When a current carrying wire is placed near the magnetometer, it produces a magnetic field opposite to the Earth's field. This new magnetic field affects the magnet's oscillations and changes its time period.

Solution:
To determine the new time period of the magnet's oscillations, we can use the concept of magnetic forces and the equation of motion.

1. Find the magnetic force:
The magnetic force on the magnet due to the Earth's field is given by the equation:
F1 = B1 * I * l * sinθ
where B1 is the strength of the Earth's magnetic field, I is the current in the wire, l is the length of the wire, and θ is the angle between the wire and the magnet.

2. Find the new magnetic force:
When the wire produces a magnetic field opposite to the Earth's field, the total magnetic force on the magnet becomes:
F_total = F1 - F2
where F2 is the magnetic force due to the new field produced by the wire. Since the magnetometer is in equilibrium during its oscillations, F_total is equal to zero.

3. Calculate the new time period:
Using the equation of motion for simple harmonic motion, we have:
F_total = -mω^2A
where m is the mass of the magnet, ω is the angular frequency, and A is the amplitude of the oscillations.

Since F_total is equal to zero, we can solve for the new angular frequency:
ω_new = sqrt((F1-F2)/(mA))

Finally, we can calculate the new time period using the equation:
T_new = 2π/ω_new

Conclusion:
By considering the magnetic forces and the equation of motion, we can determine the new time period of the magnet's oscillations when a horizontal magnetic field opposite to the Earth's field is produced by a current carrying wire.