# Currents and their Magnetic Fields

**Ampère's law**,

\[\nabla\times\mathbf{B}\left(\mathbf{r}\right)=\mu_{0}\mathbf{j}\left(\mathbf{r}\right)\]

relates the **magnetic induction** \(\mathbf{B}\left(\mathbf{r}\right)\) to the **current distribution** \(\mathbf{j}\left(\mathbf{r}\right)\).

In this section we will learn the basic techniques to calculate the magnetic induction for a given current distribution. We will use integral methods like the law of **Biot-Savart** or differential methods for certain symmetries to solve Ampère's law directly. If you have some nice problems that would fit here or you found some errors, please do not hesitate to contact us!

What happens if we take a wire, drill a **hole** in it and set some current in motion? Will we encounter a magnetic field?

We will directly use the **Biot-Savart law** to calculate the magnetic field of a thin wire via **integration**. The problem shall provide us with some intuition of Biot-Savart which we will also derive.

One of the **basic problems** of magnetostatics is the infinite wire. Using the several **symmetries** of the **current distribution** we will be able to not only find the **magnetic field** - we will also verify the "**right-hand rule**".

**Ampère's law** is hard to solve in general. What if certain **symmetries** of a current distribution are present? We may find much **simpler differential equations** which might be integrated directly. Find out what to do in **rotational** and **translational** symmetry!

**Helmholtz coils** are a devices that can provide a very **homogeneous magnetic field**. Although the principle is very old, it is still applied in **state-of-the-art experiments**. In this problem you will learn how the **special characteristics** of such coils are achieved.

In an attempt to explain the electron **spin** in a **classical model**, one can assume that it is just a rotating charged particle causing a **magnetic moment**. However, such an approach results in contradictions which renders a classical explanation of the intrinsic magnetic moment of an electron obsolete.