Electric charge, forces, fields

Reading: Chapter 21, ELECTRIC CHARGE AND ELECTRIC FIELD

Study guide: Chapter 22

Four fundamental forces

1. Gravity
2. Electric/Magnetic (note: not separate!)
3. Strong nuclear force
4. Weak nuclear force

Charge

• Like gravitational mass determines how much force it exerts on other charges, *and* how much force it feels.
• two varieties - determines whether attractive or repulsive
• quantized: units of $e=1.6\times10^{-19} $C
• conserved

Static cling

A rubbed (statically charged) balloon sticks to a wall (which hasn't been rubbed!)

We understand this in terms of an induced charge [PhET].

Coulomb's Law

In vector form

With a "source" charge $q_1$ at the origin, and a "test" charge at position $\myv r$, the force that the test charge $q_2$ feels is:

$$\myv F_e= k\frac{q_1q_2}{r^2}\uv r.$$

$$\myv F_{e1} = \left(k\frac{q_1}{r^2}\uv r\right) q = \myv E_1 q$$

So

$$\myv E_1 = \frac{\myv F_{e1}}{q_1}$$

The electrical field at a position $\myv r$ points in the same direction as the force that a positive charge at that position would feel.

Superposition

$$\begineq \myv F_\text{net} &=& \myv F_1 + \myv F_2 +\myv F_3+...\\ &=& q (\myv E_1 + \myv E_2 +\myv E_3+...) \\
&=& q\myv E_\text{net}.\endeq$$

Textbooks use as unit vectors $\bf{i}\equiv \uv x$ and $\bf{j}\equiv \uv y$ and $\bf{k}\equiv \uv z$. In the figure,

$$\begineq \myv E_1&=& 6\times 10^6 N/C \,{\bf i}+3\times 10^6 N/C\,{\bf j};\\ \myv E_2&=& 4\times 10^6 N/C \,{\bf i}-6\times 10^6 N/C\,{\bf j}.\endeq$$

The net electric field $\myv E_\text{net}$ at the blue position can be calculated from the vector components:

$$\begineq\myv E_\text{net} &=& \left[(6+4){\bf i} +(3-6){\bf j}\right]\times 10^6 N/C\\ &=& \left[10\,{\bf i} -3\,{\bf j}\right]\times 10^6 N/C.\endeq$$

Ejiri in the Suruga Province

Ejiri in the Suruga Province, by Hokusai, Katsushika. This inspired Canadian Jeff Wall: "A Sudden Gust of Wind".

Making the invisible visible

A flying carp can make the direction of the wind visible. (There's more to it than just the direction...also wind speed.)

What can we use to make the direction of the gravitational force visible?

A "plumb bob" (What's the point of doing this in golf?)

Depicting fields

Forest of vectors
direction and magnitude: vector.

Vectors

direction; arrows; magnitude: brightness.

Field lines
direction: line (tangent); magnitude: ??

Rules for drawing field lines

• Lines are tangent to direction of field.
• Number of lines $\propto$ charge.
• Lines start on + charges.
• Lines end on - charges.
• Lines never intersect (that would mean more an then one direction for the field).

$\Rightarrow |\myv E|$ is large where lines are close together, and small where lines are far apart.

And so, for an isolated charge lines should radiate equally in all directions.

Charged particles moving in electric fields

Particle moving in a uniform field

• $e=1.6\times 10^{-19}$ C
• $m_e=9.1\times 10${-31}$ kg
• $v_{0x}=10^6$ m/s
• $E=2000$ N/C

What's the electron's deviation from its straight path at the end of the field-containing region?

Use...

• Acceleration/distance formula: $y = y_0 + v_{y0} t + \frac{1}{2}at^2$:
• $t=\frac{1\ cm}{v_{0x}}$
• $a=F/m = qE/m$

...from the reading quiz

The problem with determining $\myv E$ from a measurement of the force $\myv F$ on the balloon....

If you had instead measured $\myv F$ on a 'point charge' $q$.... do you need to know the distance to the VdG to figure $\myv E$????

The strange situation of "the" Electrical field if you draw the field of two point charges. What field is the second charge reacting to???

Dipole

Dipole moment: $$\myv p = q\myv d$$

with $\myv d$ pointing from - to +.

Field perpendicular to dipole axis