**Introduction**

When the linear dimensions of a solenoid wound over a rectangular frame are increased by a factor of 3, we need to determine how the self-inductance of the solenoid changes. Self-inductance is a property of a circuit that quantifies the ability to generate an electromotive force (EMF) in response to a changing current. It depends on several factors, including the number of turns, the geometry of the solenoid, and the magnetic permeability of the material.

**Explanation**

To understand how the self-inductance changes when the linear dimensions of the solenoid are increased, we can consider the following points:

1. **Definition of self-inductance**: Self-inductance is defined as the ratio of the magnetic flux through the solenoid to the current flowing through it. It depends on the magnetic field produced by the solenoid and the geometry of the solenoid itself.

2. **Magnetic field produced by a solenoid**: The magnetic field inside a solenoid is uniform and parallel to the axis of the solenoid. It depends on the number of turns per unit length and the current flowing through the solenoid.

3. **Effect of increasing linear dimensions**: When all the linear dimensions of the solenoid are increased by a factor of 3, the volume of the solenoid increases by a factor of 3^3 = 27. This means that the number of turns per unit length remains the same, but the total number of turns increases by a factor of 27.

4. **Effect on magnetic field and magnetic flux**: Since the number of turns per unit length remains the same, the magnetic field inside the solenoid will remain the same. However, the total magnetic flux through the solenoid will increase by a factor of 27, as the total number of turns has increased.

5. **Effect on self-inductance**: Self-inductance is directly proportional to the magnetic flux through the solenoid. Therefore, when the linear dimensions of the solenoid are increased by a factor of 3, the self-inductance will also increase by a factor of 27.

**Conclusion**

When all the linear dimensions of a solenoid wound over a rectangular frame are increased by a factor of 3, the self-inductance of the solenoid increases by a factor of 27. This is because the total magnetic flux through the solenoid increases by a factor of 27, due to the increased number of turns. Self-inductance is directly proportional to the magnetic flux, so the self-inductance increases accordingly.