Magnetic Field

Recall that

• A permanent magnet creates a magnetic field at all points in the surrounding region.
• An electric current in a conductor creates a magnetic field at all points in the surrounding region.
• A moving electric charge creates a magnetic field at all points in the surrounding region.

We know

$$\vec{F}=Q(\vec{v}\times \vec{B})$$

The force on a moving charge is equal to it’s magnitude, times it’s velocity cross the magnetic field.

And we should understand that the Cross Product measures Orthogonality. Therefore if the velocity is exactly perpendicular to the magnetic field, we get the biggest number but if it’s parallel the magnetic field has no impact on the charge.

Further, the force is perpendicular to both the velocity and magnetic field!

More formulas:

$$|\vec{F}|=Q|\vec{v}||\vec{B}|\sin \theta$$ magnitude of the magnetic force acting on a moving charge $${\vec{F}}_B=I\vec{l}\times \vec{B}$$ lengths of currents in a magnetic field $$\vec{\mu }=I\vec{A}$$ magnetic moment

---

In a uniform magnetic field, a magnetic dipole (current loop) experiences:

• no net force
• torque that’s trying to align $\vec{\mu }$ and $\vec{B}$
• no torque when $\vec{\mu }$ is (anti-)parallel to $\vec{B}$
• the most torque when $\vec{\mu }$ is perpendicular to $\vec{B}$

This should feel familiar to something else…

$$\vec{\tau }=\vec{\mu }\times \vec{B}$$

---

Even more formulas…

$$\vec{B}=\frac{{\mu }_{0}q\vec{v}\times \hat{r}}{4\pi r^2}$$ magnetic field of a moving point charge $$B=\frac{{\mu }_{0}I}{2\pi r}$$ magnetic field around a current-carrying wire

Speaking of current carrying wires, did you know that for a current carrying wire, the direction of the magnet field is determined by a right-hand rule?

Grasp the wire with the thumb of your right hand in the direction of the current flow. The direction in which your fingers encircle the wire is the direction in which the magnetic field encircles the wire.

Ampère’s Law

This is used determine the formula for the magnetic field generated by a long wire.

$${\mu }_0{I}_C={\oint }_{C}B\cdot d\ell$$

Use a circle centered on the wire as the path of integration!

Biot-Savart Law

$$B=\frac{{\mu }_{0}I\sin \theta }{4\pi r^2}$$