Electrostatics

## Big Picture

All objects have positive and negative charges inside them. If the number of positive and negative charges are equal, as they most often are, then the object is neutral. Charged objects are objects with more positive charges than negative ones, or vice versa. Opposite charges attract, and similar charges repel. Electric fields are created by a net charge and point away from positive charges and towards negative charges. Many macroscopic forces can be attributed to the electrostatic forces between molecules and atoms.

## Key Terms

Charge: Charge is carried by protons and electrons. Charge is always conserved in a closed system. SI unit: C

Coulomb’s Law: Coulomb’s law states that the force between two charges is proportional to the value of the two charges and inversely proportional to the square of the distance between them.

Electric Field: Electric fields surround electrically charged particles and affect other electrically charged particles. The fields itself is a vector force fields - it indicates the direction a positive charge at a given point would move. The electric field can also be thought of as the force per unit charge at a given point, similar to a gravitational field. SI units: N/C = V/m

Conductors: Materials where electricity can pass through easily, meaning that electrons can easily move inside the material.

Insulators: Materials where electricity cannot pass through easily.

Semiconductors: Materials with a conductivity between the conductivities of insulators and normal conductors.

Superconductors: Materials with exactly zero resistance below certain temperatures.

Plasma: A conductive state of matter, similar to gas, containing ions and/or free electrons.

Voltage: Potential energy measured as a difference in potential between two points in space. SI units: V

Equipotential Lines: Lines linking points of equal voltage around a point charge. Around point charges, equipotential lines just form circles around the charge. In more complex arrangements, equipotential lines can make all sorts of shapes. Equipotential surfaces are the same as equipotential lines, except in 3-dimensional space.

## Coulomb’s Law

There are two types of charge: positive and negative.

• Electrons have negative charge.
• Protons have positive charge.
• The magnitude of the charge is the same for electrons and protons: $$e = 1.6 × 10^{-19} C$$

Coulomb’s law is used to calculate the force between two charged particles:

$$F = \frac{kq_1 q_2}{r^2}$$, where k is a constant, q is the charge, and r is the distance between the charged particles.

The force can be attractive or repulsive depending on the charges:

• Like charges (charges with the same sign) repel
• Charges with opposite signs attract

## Electric Field

Electric force can be represented by an electric field.

The field E due to a single point charge q is: $$E = \frac{kq}{r^2}$$

The field describes the force a charged object will feel if it enters the field. If another point charge q0 is in the field, it will feel a force $$F = Eq_0$$.

A simple diagram can tell us a lot about an electric field - the density of the field lines in a region indicates the strength of the field, and parallel lines (like between two charged plates) mean that the field is constant.

Electric fields can interact with each other just like gravitational fields. To determine the direction and magnitude of the electric fields at any point, just add the vectors from all of the different electric fields at that point. To the right is an illustration of the electric fields between a positive charge and a negative charge. The lines show the direction a positive charge placed in the fields will take.

Image Credit: Sharayanan, CC-BY-SA 3.0

## Physics

Electrostatics cont.

## Gauss’s Law

Gauss’s law can be used to find the electric field at any point around a charge-carrying object. Coulomb’s law can be derived by applying Gauss’s Law to a point charge. When applied to relatively simple problems, Gauss’s law essentially states that the electric field at a certain distance from a charged object is proportional to enclosed charge and inversely proportional to the area of the imaginary surface in 3-dimensional space that the electric field is passing through. This is an oversimplification of Gauss’s law. To truly understand how powerful the Gauss’s law is, you must understand a considerable amount of calculus. If you do know some calculus, the physics Gauss’s Law study guide is devoted to basic Gauss’s law problems.

## Conductive Materials

Materials can be classified as conductors and insulators.

• Metals are generally good conductors because the electrons are loosely bound to the individual atoms. There are two other classes of conductors:
• Semiconductors: Used in almost all modern electronics, semiconductors are the key component in transistors. Silicon is the most common semiconducting material.
• Superconductors: In all normal conductors, there is some resistance to the flow of electrons present. However, when some materials are cooled to very low temperatures, they will exhibit exactly zero resistance. This is called superconductivity. Superconductivity is an phenomena that can only be understood through quantum mechanics.

## Charging Methods

There are two ways to charge an object: by conduction or by induction.

Conduction is when we touch a charged object to a neutral object and the charges evenly distribute.

Charging by induction is when we charge an object without touching it. There are many methods for charging objects by induction, but here is one process for charging a single object by induction.

• First touch one finger to the neutral object to ground the object.
• Then, bring a charged object (we’ll assume it’s negatively charged, but it can be either) close to the neutral object. This causes negative charges in the neutral object to be repelled through your body to the ground.
• When the finger is removed, the neutral object will be positively charged. When charging by induction, the originally neutral object will always end up with the opposite charge.

Below is a diagram illustrating this process. When charge moves, electrons are always the ones that move. Protons cannot move between atoms.

## Physics

Electrostatics Problem Guide

## Important Equations

Note: all equations involving a charge q are only valid when q is a point charge. Calculus is required to for problems involving continuous charge distribution. Also, the value $$\frac{1}{4 \pi \varepsilon_{\circ}}$$ is often written as a constant k. However, if you pursue physics beyond an introductory level, you will learn that $$\frac{1}{4 \pi \varepsilon_{\circ}}$$ is more than just a constant and carries greater significance.

$$F = \frac{1}{4 \pi \varepsilon_{\circ}} \frac{q_1 q_2}{r^2}$$

F - force

$$ε_0$$ - permittivity of free space

q - charge

d - distance

$$F = Eq = \frac{1}{4 \pi \varepsilon_{\circ}} \frac{q}{r^2}$$

E - electric field

$$V = Er = \frac{1}{4 \pi \varepsilon_{\circ}} \frac{q}{r}$$

V - voltage

$$U_e = qV = \frac{1}{4 \pi \varepsilon_{\circ}} \frac{q_1 q_2}{r}$$

$$U_e$$ - electric potential energy

## Example Problems

Aside from the most basic electrostatics problems that involve plugging in values and basic algebra, most problems you encounter will combine electrostatics with some other area in physics. The overall strategy is to use your equations as a road map. Start with the known values and use different equations to make connections until you get to your answer.

### Example 1

One positively charged particle of charge +q and mass m1 is orbiting around a fixed, negatively charged particle of charge -q at a distance d. Determine the linear velocity of the moving particle.

### Solution

In this problem, the electrostatic force (determined by Coulomb’s law) acting between the particles is also the centripetal force keeping the positive charge in its orbit because the two particles are attracting each other. We can use this knowledge to solve for the linear velocity of the positive charge.

Tip: The negative sign on the second charge has been dropped because it is easier to do the calculations using only the absolute value of the charge. We can determine the direction of the force later on.

set centripetal force equal to Coulomb’s law

$$\frac{mv^2}{r}$$

$$=$$

$$\frac{1}{4\pi\epsilon_{circ}} \frac{q_1q_2}{r^2}$$

substitute in the given values

$$\frac{m_1v^2}{d}$$

$$=$$

$$\frac{1}{4\pi\epsilon_{circ}} \frac{q^2}{d^2}$$

solve for v

$$v$$

$$=$$

$$\sqrt{\frac{1}{4\pi\epsilon_{circ}} \frac{q^2}{d} \frac{1}{m_1}}$$

### Example 2

A charged particle with some initial velocity ($$v_0$$), charge (+q), and mass (m) is passing between two oppositely charged plates with a voltage difference (V) applied across them. The particle starts outside the plates, but let’s assume that the plates extend infinity beyond where the particle starts. A diagram is provided below. If the particle starts halfway between the plates, how far will it travel before hitting one of the plates?

### Solution

First we want to find the electric field between the plates. We know the field is constant between the plates, so we use:

$$V = Er \rightarrow E = \frac{V}{r} = \frac{v}{d}$$

We can now apply this to finding the vertical acceleration of the particle by applying F = Eq = ma.

$$Eq = ma \rightarrow \frac{V}{d}q = ma \rightarrow a = \frac{Vq}{md}$$

Now we can use the equations of kinematics to find out how long it takes for the charge to hit the plate.

$$\Delta y = \frac{1}{2}at^2 \rightarrow \frac{d}{2} = \frac{1}{2}at^2 \rightarrow t = \sqrt{\frac{d}{a}} \rightarrow t = \sqrt{\frac{md^2}{Vq}}$$

Now we can apply Δx=vt to find how far the charge will travel.

$$\Delta x = vt \rightarrow \Delta x = v_{\circ}\sqrt{\frac{md^2}{Vq}}$$

## Physics

Electrostatics Problem Guide cont.

## Charged Spheres

Whenever there are conducting metal spheres, remember that the charges are free to move within the object. Because like charges repel each other, charges tend to spread out - the charge of conducting objects is always at the surface.

If the two conducting spheres of different radii (with the same charge -q) are connected by a metal wire, the charge will ow until the electric potential of both spheres is equal.

Equation to use: $$V = \frac{1}{4 \pi \varepsilon_{\circ}} \frac{q}{r}$$

Using this equation, we can see that the charge will flow from the small sphere to the large sphere - the negative charge will flow towards the higher potential (in this case, they’re both negative, but the larger sphere’s potential is less negative).

On the other hand, charges are not free to move in nonconducting objects. Therefore, there is an even distribution of the charge density throughout the object.

## Electric Potential Energy

Equation to use: $$U_e = qV$$

If points A, B, and C represent positrons in an electric field, the field arrows represent the direction the positron will move in. Like in gravitational potential energy, where distance from the ground indicated how much potential energy an object had, the positron with the greatest distance to travel has the highest electric potential energy (in this case position A).

Keep in mind that electric potential energy (much like gravitational potential energy) does not depend on the path the object takes, rather the total distance moved!