Unit 4 - Magnetic Fields
$\vec{F_B} = q \vec{v} \times \vec{B} \rightarrow$ $ |\vec{F_B}|$ $ = qvB\sin \theta$
- Magnetic fields are made by magnetic dipoles, with magnetic north and south poles.
- Magnetic monopoles have never been found, but this does not mean that they do not exist. When a magnet splits, it forms two smaller dipoles.
- Magnetic dipoles are formed by the movement of charges.
- Ferromagnetic materials can be permanently magnetized by an external magnetic field, while paramagnetic materials are only temporarily magnetized by an external magnetic field.
- Many magnetic fields will be into or out of the page. Into the page is symbolized with an 'X', while out of the page is symbolized with a dot.
Direciton of Magnetic Field - Right-Hand Rules
Fleming's Right-Hand Rule
- Pointer finger goes in the direction of moving charge/current.
- Middle finger goes in the direction of the magnetic field.
- Thumb goes in the direction of magnetic force.
Magnetic field caused by the Current in a Wire
- If the current in a wire is going in a certain direction in a straight line, then putting your thumb in the direction of the current and allowing the rest of your fingers to curl shows the circular direction of the magnetic field.
- Conversely, if it is known that a magnetic field line is going in a certain direction in a straight line, then putting your thumb in the direction of the magnetic field line and allowing the rest of your fingers to curl shows the circular direction of the current.
Magnetic Force on Current
$\vec{F_B} = I \vec{L} \times \vec{B} = \int I \, (d \vec{L} \times \vec{B}) \rightarrow$ $|\vec{F_B}|$ $= ILB \sin \theta$
- You can still use Fleming's Right-Hand Rule!!
Ampère's Law
$\oint B \, d \ell = \mu_0 I_{enc}$
- This equation is useful for trying to find the magnetic field of a current-carrying wire.
- This equation makes use of the loop integral, not the surface integral
- Using this equation, it can be found that for a current-carrying wire, it can be found that $B = \frac{\mu_0 I}{2 \pi r}$.
Biot-Savart Law
$d\vec{B} = \frac{\mu_0}{4\pi} \, \frac{Id\vec{L} \times \hat{r}}{r^2}$
- This equation was experimentally derived; it was not derived mathematically.
- This equation is overly complicated to use, so try to avoid it if possible by using Ampère's Law.
- However, there is one situation where it is necessary: finding the magnetic field from a circular current-carrying wire.
- Using this equation, it can be found that for a current-carrying wire, it can be found that $B = \frac{\mu_0 I}{2 \pi r}$.
Mass Spectrometer
Cross Field in Velocity Selector
- In the first section of a mass spectrometer, the electric field and uniform magnetic field are crossed in such a way that $\vec{F_E}$ and $\vec{F_B}$ cancel each other out.
- This leaves the velocity of a particle passing through this section as $v = \frac{E}{B}$
Deflection Chamber
$\frac{m}{q} = \frac{RB^2}{E}$
- Only the uniform magnetic field is present in the deflection chamber. As such, only $\vec{F_B}$ acts on the particle.
- Each particle will move across the chamber in a circular motion with radius $R$.
- Positive charges deflect upwards in a normal crossed field, while negative charges deflect downwards.