Electrostatics Notes

Charging by Conduction

To charge an electroscope by conduction, a charged metal rod touches the electroscope's knob.
Bringing a negatively charged rod near a negatively charged electroscope causes the leaves to spread apart farther.
Bringing a positively charged rod near a negatively charged electroscope causes the leaves to fall closer together.

Coulomb's Law

Coulomb (1770s) studied the force of repulsion between charged spheres.
- The magnitude of the force ( $F$ ) varies inversely with the square of the distance ( $r$ ) between the centers of the spheres: $F \propto \frac{1}{r^2}$ .
- The force varies directly with the charge of the bodies ( $qA$ and $qB$ ): $F \propto qA qB$ .
- Combining these proportionalities:
 $F \propto \frac{qA qB}{r^2}$
Coulomb's Law states that the force between two charges is equal to a constant ( $K$ ) times the product of the two charges divided by the square of the distance between them: $F = K \frac{qA qB}{r^2}$
- When charges are measured in Coulombs, distance in meters, and force in Newtons, the constant $K$ is approximately $9.0 \times 10^9 \frac{N \cdot m^2}{C^2}$ .
- Coulomb's Law is valid only for uniform charges.

Electroscope

An electroscope consists of a metal knob connected by a metal stem to two thin, lightweight pieces of metal called leaves, enclosed to eliminate air currents.
The metal foil leaves on a neutral electroscope hang loosely together.

Vocabulary

Electrostatics: The study of electric charges that can be collected and held in one place.
Neutral: When the amount of negative charge in an object is equal to its positive charge.
Insulator: A material in which charge doesn't move through easily.
Conductor: A material that allows charges to move about easily.
Electroscope: A device used to measure charge, consisting of a metal knob connected by a metal stem to two thin, lightweight pieces of metal leaves, enclosed to eliminate air currents.
Charging by Conduction: Charging a neutral object by touching it with a charged object.
Charging by Induction: Charging a neutral object by bringing a charged object near it.
Grounding: The process of removing excess charge by connecting an object to the Earth.
Coulomb's Law: The magnitude of the force between point charge $qA$ and point charge $qB$ , separated by distance $r$ , is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
Coulomb: The SI unit for charge, equivalent to the charge of $6.24 \times 10^{18}$ protons or electrons.
Elementary Charge: The magnitude of the charge of a single electron or proton ( $1.602 \times 10^{-19} C$ ).

Materials and Charge

Ben Franklin (1750s) noted that materials have varying degrees of ability to acquire charge.
- Hard rubber and plastic tend to become negatively charged.
- Glass and wool tend to become positively charged.
Metal is a good conductor because electrons can move freely in it.
Plasma and graphite are also good conductors.

Electric Field Lines

Single Charge: For a single charge, the electric field lines radiate from that charge.
- Field lines radiate outward from a positive charge.
- Field lines radiate inward toward a negative charge.
- The direction of the electric field ( $E$ ) at any point is tangent to the field line.
Two Equal and Unlike Charges: The electric field lines of two equal and unlike charges form continuous lines from the positive charge to the negative charge.
- The vector at point P shows the direction of the electric field ( $E$ ) at point P.
- Where the lines are closest, a charged particle would experience the most force.
Two Like Charges: The electric field lines around two like charges never connect with each other.
- The vector at point P shows the direction of the electric field ( $E$ ) at point P.
- A charge would experience no electrostatic force in the center where there are no electric field lines.

Calculating Electric Field

If and only if the charge $q'$ is a point charge or a uniformly charged sphere, you can calculate its electric field from Coulomb's Law:
$E = \frac{F_{on q'}}{q'} = \frac{Kq q'}{r^2} \times \frac{1}{q'} = \frac{Kq}{r^2}$
The magnitude of the force depends on the magnitude of the electric field ( $E$ ) and the magnitude of the charge ( $q$ ):
$F = Eq$
An electric field is stronger where lines are closer together and weaker where lines are spaced out.

Vocabulary

Electric Field: A property of the space around a charged object that exerts force on other charged objects.
Electric Field Line: Indicates the direction of the force due to the electric field on a positive test charge.

Electric Field Measurement

The strength of an electric field is equal to the force on a positive test charge divided by the strength of the test charge:
$E = \frac{F_{on q'}}{q'}$
The magnitude of the electric field is measured in Newtons per Coulomb (N/C).

Electric Potential Inside Conductors

When charged, the electric potential difference between any two points inside a closed metal container is zero.
The electric field outside the container is not zero; it is always perpendicular to the surface of the conductor.

Capacitors

Manufacturers make capacitors from two conducting plates separated by a thin layer of insulator.
These are often rolled into cylinders.
Engineers design different capacitors to have specific capacitances.
Capacitance: The ratio of the magnitude of the net charge on one plate of the capacitor to the potential difference across the plates.
$C = \frac{q}{V}$
Capacitance is measured in farads (F), where 1 F = 1 C/V.

Charge Distribution

Irregular Surface: On an irregular conducting surface, the charges are closest together at sharp points.
Hollow Sphere: The charges on the hollow sphere are entirely on the outer surface.
Conducting Sphere: On a conducting sphere, the charge is evenly distributed around the surface.

Electric Potential Difference in a Uniform Field

Electric Potential Difference in a Uniform field: $\Delta V = Ed$
- The potential difference between two locations in a uniform electric field equals the product of the electric field intensity and the distance between the locations parallel to the direction of the field.

Electric Potential

The electric potential difference from point A to point B:
$\Delta V = VB - VA$
Electric potential differences are measured with a voltmeter.
Electric potential difference is sometimes called voltage (do not confuse with Volts (V)).
Positive Electric Potential Difference:
$PE(A) + W_{on charge}^{by you} = PE(B)$
Negative Electric Potential Difference:
$PE(B) + W_{on charge}^{by charge} = PE(A)$

Vocabulary

Electric Potential Difference: The work ( $W_{on q'}$ ) needed to move a positive test charge from one point to another, divided by the magnitude of the test charge.
Volt: One joule per one Coulomb.
Equipotential: When the electric potential difference between two or more positions is zero.
Capacitor: Modern device for storing electrical energy.
Capacitance: The slope of the line in a charge vs. potential difference graph.

Electric Fields and Potential Energy

Both the gravitational force ( $Fg$ ) and the gravitational field ( $g = \frac{Fg}{m}$ ) point toward Earth.
If you do work on a ball to lift it away from Earth's surface, the gravitational potential energy in the ball-Earth system increases.
The same is true for the electric field. The work done to move a charged particle in an electric field can result in the particle gaining electrical potential energy or kinetic energy, or both.
This is basically the same with two unlike charges: doing work to separate the charges increases the electrical potential energy of the system.
The longer the charge or distance moved, the greater the increase in electric potential energy ( $\Delta PE$ ).
To find the work done to move a charge, use the equation:

Resistance

$R = \frac{\Delta V}{I}$

Measured in ohms ( $\Omega$ ), one ohm is the electric charge of 1 Ampere being permitted to flow when a potential difference of 1 Volt is applied across the resistance.