# Fields

Non-contact forces:
- strong nuclear force
- weak nuclear force
- electromagnetic force
- - electric fields (dipoles & monopoles)
- - magnetic fields (dipoles only)
- gravitational force (monopoles only)

Gravitational & -ve electric monopoles - field lines radiate towards central object
Magnetic & electric dipoles - field lines go from + to -, or N to S

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## Gravity

### Newton's law of universal gravitation

$$F_g=G{{m_1m_2}\over r^2}$$

where
$F_g$ is the gravitational force between $m_1$ and $m_2$
$G$ is the gravitational constant, $6.67 \times 10^{-11} \operatorname{N m^2kg^{-2}}$
$r$ is the distance between centre of $m_1$ and $m_2$


- inverse square law
- acceleration can be calculated from $F_g$, since $F=ma$
- all objects with mass attract each other with $F_g$
- $F_g$ acts equally on $m_1$ and $m_2$
- acceleration of an object close to earth's surface can be approximated by ignoring its mass ($m_2 \approx 0$)
- apparent weight may be different to gravitational (normal) weight

### Gravitational fields

$$g={F_g \over m}=G{M \over r^2}$$

where
$g$ is the gravitational field strength
$F_g$ is the force due to gravity ($=G{{m_1m_2}\over r^2}$)
$m$ is the mass of object in the field
$M$ is the mass of the central body

- arrows towards centre of object
- closer arrows mean larger force
- parallel field lines - uniform field strength (vector)

Characteristics of gravitational fields:
- monopoles
- attractive force
- extends to infinite distance, but diminishes with inverse square law
- charge produced by gravity = $GM$

### Work in a gravitational field

Gravitational potential energy: $E_g = mg \Delta h$
Work: $W = \Delta E_g = Fx$

Area under force-distance graph = $\Delta E_g$
Area under field-distance graph = $\Delta E_g / \operatorname{kg}$

### Satellites

## Magnetic fields

### Characteristics
- field lines always go from N -> S
- dot means out of page, cross means into page
- ${E_1 \over E_2}={r_1 \over r_2}^2$
- flux: change in magnetic field


## Electric fields

### Characterisics

- surrounds +ve and -ve charges
- exerts force on other changes in its field
- monopoles and dipoles
- attractive/repulsive forces
- can be constrained to a fixed distance (conductors / insulators)
- current flows from +ve to -ve

### Field lines
- +ve to -ve
- start and end $\perp$ to surface
- field lines never cross
- point charges - radiate from centre

### Forces

$$F=qE$$

where
$F$ is the force on charged particle
$q$ is the charge of object experiencing force (Coulombs)
$E$ is the strength of the electric field (Newtons / Coloumb or Volts / metre)

### Work in electric fields

$$W=qV$$

where
$W$ is the work done on +ve point charge or in field
$q$ is the charge of point charge being acted on
$V$ is the potential (voltage) between points

### Coulomb's law


$$F=k{{q_1q_2}\over r^2}$$

where
$k$ is Coulomb's constant $9.0 \times 10^9 \operatorname{N m^2 C^{-2}}$
$q_1$ and $q_2$ are the charges on the interacting points


### Electric field at distance from a charge

$$E=k{Q \over r^2}$$

### Lenz's law
- Right hand grip rule (relationship between directions of $I, F$)
- Eddy currents counter movement within a field
- Represented by -ve sign in Faraday's law

### Solenoids
- Coil around core (like a transformer but field is transferred to kinetic energy)

### Magnetic force on charged particles

$$F=qvB$$

where
$v$ is the component of velocity which is $\perp$ to magnetic field

### Right hand slap rule


**Field, current and force are all 90 degree to each other**
<pre>
force
|      /    field
|   /
|/  90 de=
 \
   \   +ve charge
</pre>

Force is given by $F=nBIl$


### Faraday's law of induction

$$\epsilon = -N{{\Delta \Phi_B}\over{\Delta t}}$$

where
$\epsilon$ is induced EMF (voltage)
$N$ is the number of turns in the primary coil
$\Phi_B$ is the magnetic flux (Wb or V / s)
$\Delta t$ is the change in time for one cycle (can be derived from period or frequency)

### Flux through coils
$$\Phi_B = B_{\perp}A$$

where
$B_\perp$ is the field strength (Tesla)
$A$ is the area of the field perpendicular to field lines

if $B {\not \perp} A, \Phi_B \rightarrow 0$
if $B \parallel A, \Phi_B = 0$

- flux-time graphs ($t$ on $x$-axis): $\operatorname{gradient} \times n = \operatorname{emf}$


**EMF is proportionate to change in flux**

**Induced EMF opposes (counters) change in flux**

### Transformer equation

$${V_p \over V_s}={N_p \over N_s}$$
$${I_p \over I_s}={N_s \over N_p}$$

- core strengthens and "focuses" ac flux $\Phi$ through secondary coil


### Root mean square

$$V_{\operatorname{rms}} = {V_{\operatorname{p\rightarrow p}} \over \sqrt{2}}$$

## Power transmission
- 240 V / 50 Hz in Australia
- higher voltages have lower $V_{\operatorname{loss}}$
- ac is used because its voltage is easily changed with xfmrs

### Safety
- $\ge 30 \operatorname{mA}$ through heart is dangerous

### Transmission $P_{\operatorname{loss}}$

$$P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R}$$

where
$R$ is the total resistance (derived from resistance per distance)

To reduce power loss, use lower resistance (thicker) wires or increase voltage / reduce current with transformers



### Motors

#### DC

- current-carrying wire experiences magnetic force $F$ equal to $nBIl$
- torque: $\tau = r_{\perp} F$
- split ring and brushes