Magnetism

Magnetic Field

A moving charge causes a magnetic field, following the right hand rule: Your thumb pointing towards the direction of movement of the positive charge, and your other fingers wrap around to indicate the direction of the magnetic field.

A circulating current forms a magnetic dipole.

Calculate Field

Magnetic field of a current with right hand rule

${\vec {B}}({\vec {r}})={\frac {\mu _{0}}{4\pi }}{\frac {I_{1}d{\vec {l}}_{1}\times {\hat {r}}}{r^{2}}}={\frac {\mu _{0}}{4\pi }}{\frac {\left|Idl\right|}{r^{3}}}\rho {\hat {e}}_{\phi }$

Where $\mu _{0}$ is magnetic permeability, and $\rho$ is distance from the wire.

Any component going along the direction of current is cancelled by cross product. Something else. Therefore, it is unsurprising that magnetic field is circulating.

Straight Line

Consider a straight line segment starting at $z_{1}$and stopping at $z_{2}$

${\vec {B}}={\frac {\mu _{0}I}{4\pi \rho }}\left({\frac {z_{2}}{\sqrt {z_{2}^{2}+\rho ^{2}}}}-{\frac {z_{1}}{\sqrt {z_{1}^{2}+\rho ^{2}}}}\right){\hat {e_{\phi }}}$

Circular Loops

Let there be a loop of current flowing clockwise. The magnetic field inside the loop is always going out of the page, whereas the field outside always go into the page.

This can be proven by considering each single length of wire. The shape of the loop does not matter.

A pair of circular loops can produce a very uniform magnetic field in the middle.

Solenoid

Solenoids use this property to create uniform magnetic fields. They are essentially a long coil of current. Inside the solenoid there is a strong, uniform magnetic field. At the ends, the strength drop drastically.

The field created can be derived from adding together circular loops.

${\vec {B}}_{inside,axis}=\mu _{0}{\frac {N}{L}}I{\hat {k}}$

Toroidal Solenoid

What if we wrap around the solenoid so that they form a loop of coil? we get a circulating magnetic field. The magnitude is the same.

Rings

Consider a hollow inner conducting shell inside a hollow outer conducting shell. Run current through the shells and you get a ring of magnetic field.

Ampere's Law of Magnetic field

$\oint _{C}{\vec {B}}\cdot d{\vec {l}}=\mu _{0}\iint _{S_{L}}{\vec {J}}\cdot d{\vec {s}}=\mu _{0}I$

The general proof of Ampere's Law involves starting from Biot-Savart Law, integrating to a term and a Maxwell's term. I will do the proof later.