Magnetic fields are really important when it comes to superconductivity. Superconductivity is when materials can carry electricity without any resistance, and this happens below a specific temperature called the critical temperature, or $T_c$. Understanding how magnetic fields affect superconductivity can be broken down into two main ideas: how outside magnetic fields influence materials and how the materials’ own magnetic properties behave.

First, let's look at how superconductors react to external magnetic fields. There are two main types of superconductors: Type I and Type II.

Type I superconductors completely push out magnetic fields until they reach a certain strength called the critical field strength, or $H_c$. If the magnetic field gets too strong, these superconductors lose their special properties and act like regular conductors. You can see this behavior in pure materials like lead and mercury.

On the other hand, Type II superconductors, which are often high-temperature superconductors, have a more complicated relationship with magnetic fields. These materials allow some magnetism to enter them in a specific way, creating tiny whirlpool-like areas called vortices. In this state, both superconductivity and magnetism can exist together, but only up to two different strengths: $H_{c1}$ and $H_{c2}$.

Now, a key factor in how well superconductors work in magnetic fields is the critical current density, or $J_c$. This measures the most current a superconductor can handle without losing its superconducting state because of heat or magnetic issues. How well a superconductor can hold onto the magnetic lines also affects $J_c$. Improving the locking of these lines can be done by adding some defects, using different materials, or creating mixed structures. This can make a superconductor work much better in a magnetic field.

Magnetic flux pinning, or locking magnetic lines in place, is also important for practical uses. In technologies like magnetic levitation, MRI machines, and particle accelerators, it's crucial for superconductors to keep their special properties even when magnetic fields are at play. Type II superconductors can handle this well and keep a steady superconducting state, which is key for high-performance superconducting magnets that work in strong magnetic field environments.

The magnetic properties of materials also impact how they behave as superconductors. Some materials have tiny magnetic moments that can interfere with the formation of Cooper pairs, which are pairs of electrons that create superconductivity according to the Bardeen-Cooper-Schrieffer (BCS) theory. In certain magnetic materials, these magnetic moments can fight against the attractive forces needed for superconductivity, and this is particularly noticeable in heavy fermion materials and iron-based superconductors.

One fascinating idea is how superconductivity and magnetism can work together. There are materials that show both superconductivity and ferromagnetism, leading to interesting phases where both can exist. Sometimes, magnetism can even help superconductivity, while in other situations, they can limit each other.

High-temperature superconductors, particularly those discovered in the 1980s, are very sensitive to magnetic fields. Researchers are still looking into how the magnetic order in these materials affects superconductivity. For example, the copper oxide planes in copper oxide superconductors display magnetic changes that influence how electrons pair up. Scientists can tweak the critical temperature $T_c$ by adding magnetic materials or controlling outside magnetic fields, which helps them understand superconductivity in these complex materials.

When it comes to real-world uses, there’s a push to create new materials that have excellent superconducting abilities while also being strong in magnetic environments. Right now, scientists are experimenting with different chemical mixtures and structures. For example, iron pnictides and chalcogenides show strong superconductivity and good responses to magnetic fields because of their layered designs.

In industries, designing superconducting wires and tapes for power use, like for power lines and limits on electrical faults, needs careful thinking about magnetic environments. Coated conductors, which have a thin layer of high-temperature superconductor on a metal base, aim to work well even in strong magnetic fields. Knowing how magnetic fields affect superconductivity is key for designing and making these materials.

The effects of magnetic fields are also crucial for quantum computing. Superconducting qubits, which are parts of many quantum computers, are made from superconducting circuits that depend on special connections called Josephson junctions. The performance of these qubits can change with magnetic flux, affecting how well they work. To reduce these magnetic effects, scientists carefully design how the qubits and materials are made.

Looking to the future, understanding how magnetic fields affect new materials will be very important in superconductor research. Current studies are diving into complex superconductors, especially in quantum materials, which may reveal new ways superconductivity can work and its potential uses.

In summary, magnetic fields significantly influence superconductivity in many ways. The interactions between outside magnetic fields, the materials’ own magnetic properties, and how electrons pair together decide if a material can keep its superconducting features or becomes unstable. This complex relationship not only helps us understand superconductors better but also shapes how we use superconducting technologies in various fields, from energy to quantum computing. As research continues, the link between magnetic fields and superconductivity will keep evolving, leading to exciting discoveries.