Students will learn the idea of magnetic field lines, how they behave in the situation of permanent magnets and current carrying wires and also how to calculate the magnetic field at an arbitrary distance from the wire.

### Key Equations

\begin{align*}B_{\text{wire}} = \frac{\mu_0 I}{2 \pi r}&& \text{Magnetic field at a distance } r \text{ from a current-carrying wire}\\ \text{Where } \mu_0=4 \pi \times 10^{-7} \; \text{Tm/A} && \text{Permeability of Vacuum (approximately same for air also)} \end{align*}

### Guidance

Permanent magnets (like refrigerator magnets) consist of atoms, such as iron, for which the magnetic moments (roughly electron spin) of the electrons are “lined up” all across the atom. This means that their magnetic fields add up, rather than canceling each other out. The net effect is noticeable because so many atoms have lined up. The magnetic field of such a magnet always points from the north pole to the south. The magnetic field of a bar magnet, for example, is illustrated below:

If we were to cut the magnet above in half, it would still have north and south poles; the resulting magnetic field would be qualitatively the same as the one above (but weaker).

Charged particles in motion also generate magnetic fields. The most frequently used example is a current carrying wire, since current is literally moving charged particles. The magnitude of a field generated by a wire depends on distance to the wire and strength of the current \begin{align*}(I) \end{align*} (see 'Key Equations' section) :

Meanwhile, its direction can be found using the so called **first right hand rule**: point your thumb in the direction of the current. Then, curl your fingers around the wire. The direction your fingers will point in the same direction as the field. Be sure to use your right hand!

Sometimes, it is necessary to represent such three dimensional fields on a two dimensional sheet of paper. The following example illustrates how this is done.

In the example above, a current is running along a wire towards the top of your page. The magnetic field is circling the current carrying wire in loops which are perpendicular to the page. Where these loops intersect this piece of paper, *we use the symbol \begin{align*}\bigodot\end{align*} to represent where the magnetic field is coming* *out of the page**and the symbol \begin{align*}\bigotimes\end{align*} to represent where the magnetic field is going* *into the page**.* This convention can be used for all vector quantities: fields, forces, velocities, etc.

#### Example 1

You are standing right next to a current carrying wire and decide to throw your magnetic field sensor some distance perpendicular to the wire. When you go to retrieve your sensor, it shows the magnetic field where it landed to be \begin{align*}4*10^{-5}\;\text{T}\end{align*}. If you know the wire was carrying 300A, how far did you throw the sensor?

##### Solution

To solve this problem, we will just use the equation given above and solve for the radius.

\begin{align*} B&=\frac{\mu_oI}{2\pi r}\\ r&=\frac{\mu_oI}{2\pi B}\\ r&=\frac{4\pi*10^{-7}\:\text{Tm/A}* 300\:\text{A}}{2\pi* 4*10^{-5}\:\text{T}}\\ r&=1.5\:\text{m}\\ \end{align*}

### Watch this Explanation

### Simulation

Magnet and Compass (PhET Simulation)

### Time for Practice

- Sketch the magnetic field lines for the horseshoe magnet shown here. Then, show the direction in which the two compasses (shown as circles) should point considering their positions. In other words, draw an arrow in the compass that represents North in relation to the compass magnet.
- Find the magnetic field a distance of 20 cm from a wire that is carrying 3 A of electrical current.
- In order to measure the current from big power lines the worker simply clamps a device around the wire. This provides safety when dealing with such high currents. The worker simply measures the magnetic field and deduces the current using the laws of physics. Let's say a worker uses such a clamp and the device registers a magnetic field of 0.02 T. The clamp is 0.05 m from the wire. What is the electrical current in the wire?

#### Answers to Selected Problems

- Both pointing away from north
- \begin{align*} 3 \times 10^{-6} \text{T} \end{align*}
- 5000 A