Electric Charge Practice Problems

 

Operational Definitions

An operational definition is a simple procedure that any kindergartner can follow which allows us to easily classify things. Creating operational definitions is oftentimes the first step in building a mental model for something complicated. For example to operational define "flammable" you would tell someone to:

 

1. Put the object in an open flame. 
2. Notice if the object catches fire. 
3. If the object catches fire, it is flammable. If the object does not catch fire it isn't flammable.  

 

Notice that anyone could easily use this definition to classify objects as either flammable or non-flammable. However operational definitions don't give us any theoretical insight. In this case even if we follow the procedure we don't have any idea why certain things catch on fire but other materials don't. 

 

Operational Definition to classify an object as "charged" or "uncharged:"

 

     1. Take an object and hold it above neutral paper dots.

     2. If the object interacts with the paper dots (i.e. the dots flutter around/stick to the object) then the object is charged.

     3. If the object does not interact with the paper dots, then it is not charged.

 

Tip: You use this same operational definition to show that paper dots are neutral because they don't attract other paper dots. 

 

Cornerstones of Electric Charge

The Electric Charge Cornerstones act as your foundation for understanding Electric Charge. All of these rules were created empirically (i.e. via experimental evidence). The evidence for each rule is listed below. 

 

1. There are only two types of charge (+ and -)
2. Opposite charges attract each other
3. Like charges repel each other
4. Electric charge is conserved: electric charge cannot be created/destroyed
5. Neutral charges are attracted to both positive and negative charges
6. Neutral objects contain an equal amount of positive and negative charges - they do not have any excess charges within them
7. The closer two charges are to each other, the stronger their interaction (either attraction or repulsion)

 

Evidence for the Cornerstones:   Like charges repel- We know this is true because if we put two charged objects (i.e. tapes) near to each other, they move away from each other. We know they are charged because each object individually attracts the paper dots which are neutral (they don't attract other paper dots). Moreover, we know the objects have like charges because we created the tapes in the same way.    Opposite charges attract- We know this is true because if we put a top and bottom tape near to each other (again, we know they are charged because they attract the paper dots), they go near to each other or attract. We know they are opposite charges as due to our last cornerstone, we know that they cannot be like charges as they do not repel; therefore, they must be opposite.   The closer two charges are to each other, the stronger their interaction- (either attraction or repulsion). We know this cornerstone is true because if we put two charged objects near each other (again, we know they are charged because they attract the paper dots), the closer they are the stronger the attraction. For example, two charged pieces of tape will deflect more (either attract more or repel more) when closer compared to if they were further apart.    Neutral and charged objects attract- We know an object is neutral because it neither attracts nor repels the paper dots (it has no interaction). When you put a neutrally charged object near a charged object, there is an attraction. For example the charged tape attracts the neutral paper dots. We know they attract because they move closer to each other.   Neutral contains equal amounts of opposite charge- the evidence for this cornerstone is found in the experiment which started with a top and bottom tape together, then the two tapes were separated, and then put back together again. When the top and bottom tape are together we know that they are neutral since the combined top/bottom tape does not interact with the neutral paper dots. When we separated the top and bottom tape both tapes ended up oppositely charged- we knew they were charged because they each attracted the paper dots. We know they are not the same charge because they attract each other. Once we put the tapes back together, they became neutral again as together they neither attracted nor repelled the paper dots. There are only two possible explanations to explain this. (1) the charge poofed into existence when the charges were separated and poofed out of existence when recombined or (2) the charge was there the whole time, we just didn't notice it when there was an equal amount of positive and negative. We noticed the charge when the tapes were separated and had more positive then negative (or visa versa), but didn't notice the charge when there were equal amounts of positive and negative. Argument (2) is much more persuasive and although we are not able to prove it with the tools in a 9th grade physics lab, it has been experimentally supported elsewhere.    Electric charge is conserved- it cannot be created nor destroyed. The last experiment also supplies evidence for this cornerstone. Throughout the whole process of the combined tapes, separating them, and then recombining them again the total amount of charge stays the same. If this were not the case, the recombined tapes should not be neutral again.

 

Insulators & Conductors

 

Operational Definition for Insulators and Conductors

 

1. Hold the object in your bare hand. 
2. Rub object with fur. 
3. While still holding the object wave it over paper dots. 
4. If the paper dots attract to the object, the object is an insulator.
5. If the object does not react at all to the paper dots then it is a conductor.

 

Tip: this operational definition is very similar to the operational definition to determine if an object is charged or not. However in this definition you rub the object with the fur, but in the previous operational definition you don't! Try not to mix them up!  

 

Common Insulators

PVC (plastic)

glass

most rubbers

dry air

styrofoam

wax

 

Common Conductors

metals

people (i.e. human body)

Earth

wood

graphite

paper

 

Theoretical Definition of Insulators and Conductors

 

Conductors are different than insulators based on how freely electric charge can move around. In a conductor the charge is free to move anywhere. However in an insulator the charges can only move short distances inside little spheres (actually atoms). 

 

Insulator: charge free to move inside the bubble, but not outside/between bubbles.

Conductor: charge free to move anywhere.

 

Tip: notice the charge gradient in the conductor. Most of the excess positive charges move as close as possible to the negative rod, while most of the excess negative charges move away. In the insulator, however, the excess positive charges still move towards the negative rod, but they do not move outside of their respective "bubbles".

 

Evidence for the Theoretical Definition of Insulators and Conductors

 

In the Conductors and Insulators tutorial, we imagined what would happen if we cut a rod in half while it was near another charged object. We considered two possibilities: (1) what if the charge can only move in small regions and (2) what if the charge can move anywhere. If you cut the red rod (above) in half, assuming that you didn't slice a "bubble" (atom) in half, each side would still be neutral since it would have the same amount of excess positive as negative. If you brought one of these red halves near the paper dots nothing would happen since neutral does not interact with neutral objects. When we actually did this test, the plastic rod behaved this way (i.e. each half was neutral after the cut). However if you cut the green object in half and then either half brought near the paper dots it would attract them since each half would be charged. The metal rod acted like this. This provides strong evidence that charge can only move in small regions in an insulator but can move anywhere within a conductor. 

 

Tip: we know that charge can move a little within insulators because any charged object always attracted to neutral objects whether conductors or insulators. The reason the charged object will attract a neutral insulator is that some opposite excess charge moves a little closer to the charged object while some excess like charge moves a little further way (since opposite charges attract and like charges repel). Even though there are as many attractive forces as repulsive ones, the opposite charges are a little closer together so the attractive forces are a little stronger. This results in a net attraction between the charged object and the neutral insulator. A similar thing happens with the conductor except that the charges in the neutral conductor spread apart much more. 

 

How can we use the Theoretical Difference between Insulators and Conductors to Explain the Operational Definition?

 

The reason that these operational definitions work is because if the object is an insulator, the excess charge is not free to move within the material. When you rub a plastic rod with fur rubbing the two materials together causes excess charge to separate. For example, the fur may become excess positive while the rod becomes equally excess negative. However once this excess negative charge is moved onto the rod it is not free to move large distances (it's basically stuck where you put it). So the plastic rod is now charged and will attract to the neutral dots. 

 

The process with the metal rod starts out similar. When the fur is rubbed on the metal rod excess charges once again separate. Let's say the rod gets some excess negative charge while the fur becomes excess positively charged. Keep in mind that the rod, the hand holding the rod, and the Earth are all conductors. Based on our theoretical definition we know that excess charge can move freely within any conductors which are touching. This means the excess charge can move between the rod, person, and Earth. So even though some excess negative charge is initially transferred to the metal rod it does not stay there. Like charges repel and the excess charges end up in the Earth where they can spread out the most (i.e. be most stable). When you hold the metal rod over the neutral dots the rod don't interact because the metal rod is no longer charged. Essentially the metal rod can be grounded while the plastic one isn't. 

 

Tip: whether an object is a conductor or insulator is totally separate from if the object is charged or not. Just because you can't charge a conductor when it's held in your bare hand doesn't mean that it can never be charged! For example, the metal rod in the Conductors and Insulators tutorial ended up charged when it was "cut in half" near a charged object. 

 

Electroscopes

 

We can use an electroscope to determine if an object is charged or not. The picture below shows an electroscope with a net negative charge brought near a negatively charged rod. 

 

What are the basic parts of electroscopes & how does it work?   Metal leaf electroscopes have two leaves of metal foil on the bottom and a tall rod on the top. Using an electroscope, it is easy to tell whether or not an object is charged as the leaves will repel if the electroscope gains a net charge.   If you place a charged rod near, but not touching a neutral electroscope you will notice the electroscopes leaves repel as long as the charged object is close by. Once the charged object is removed the neutral electroscope leaves will hang straight down.  The leaves of an electroscope repel if the have the same net charge (like charges repel).  Although the top and the bottom of an electroscope may have a different net charge, the leaves will always either be neutral or the same net charge.

 

 

Electric Charge Distribution Diagrams Approach

 

Step 1: Define the System Step 2: Determine the net charge Step 3: Are the charges mobile (i.e. are the materials insulators or conductors?)? Step 4: Do any charges move in or out of the system? If not, be sure to show conservation of charge.  Step 5: Figure out the most stable configuration. Like charges are most stable when they are as far apart as possible. Opposite charges are as stable as possible when they are as close together as possible.

 

What does "Excess Charge" Mean?

 

In charge distribution diagrams, a sign does not mean one single charged particle. A sign means that that specific area has an excess charge with that sign. There are billions of particles per sign. Excess charge is basically net charge for this region. For example, if there are more positive charges on an object than negative, the excess charge is positive, but there are still negative charges. A neutral object should have equal numbers of both charges, and therefore it does not have any excess charge. 

 

Wrong way to draw a neutral object:   There shouldn't ever be excess positive at the same spot as excess negative in a conductor. This is a logical contradiction since you can't have more negative then positive charges and also more positive then negative charges at the same spot.

Correct way to draw a neutral object:     This is the correct way to draw a neutral object. There are no extra positive charges and no extra negative charges. Note: we are NOT saying that there isn't any charge. There are equal amounts of opposite charges (see neutral object cornerstone).

 

Example 1:

A positively charged Van de Graaff generator is near a neutral pie tin. Draw the charge distribution for the pie tin. 

 

What if a hand touches the pie tin. Redraw the charge distribution.  

 

This last case where the hand touched the neutral pie tin and ended up giving it an excess negative charge is an example of charging by induction. Note that the hand acted as a ground. 

 

 

Grounding

 

The three qualities which make a good ground are:

1. Large (compared to the object which is being grounded, say 1000 times the volume)
2. Neutral
3. Conductor

 

Many times, touching a conductor to a ground will neutralize it. Since like charges spread out as far as possible, if a negatively charged metal sphere is touched to the Earth all the excess charge will move into the Earth (since it can spread out way more there). This leaves the metal sphere neutral (since the Earth is so much bigger then the sphere that the percent of excess charge remaining in the sphere is negligible). 

 

Tip: be wary: grounding does not always neutralize a conductor. For example, charging by induction actually charges an object through a ground. 

 

 

Charging by Conduction, Induction, and Friction

 

The three ways to charge an object are:

     1. Conduction - when charges transfer between two conductors which are touching. 

     2. Induction - an indirect method of charging using a ground

     3. Friction - rub two objects together (like the fur and PVC pipe). Charges on the fur/PVC will separate. One will become excess positive while the other becomes equally excess negative. 

 

Charging by Conduction

 

One way to charge a conductor is to charge it through conduction. To charge by conduction you must have the two conducting objects touching. Since like charges repel, like charges will spread out as far as possible to be as stable as possible. Any opposite excess charges will meet up and cancel out (one "+" and one "-" is simply neutral, which we show by leaving the area blank). As the excess charges move around between the two conductors it can change the net charge of each object which is what we call "charging by conduction."

 

Charging by Induction

 

In math, a proof by induction is an indirect way of showing why something is true. Charging by induction is a much less direct way of charging a conductor compared to charging by conduction. Let's imagine we have a neutral metal rod we want to charge excess negative by induction. First bring a another positively charged rod (let's say it's plastic), near but not touching the metal rod. Then ground the metal rod by connecting it to something like the Earth with a wire. The Earth is neutral so it has many positive and negative charges, though it is neutral overall. Since the metal rod (conductor) is touching the Earth (conductor), charges can freely move between them. With a positively charged plastic rod near, but not touching, the metal rod, excess negative charges from the Earth will be attracted up into the metal rod (opposite charges attract). If we want to permanently charge the metal rod it's important we first remove the grounding wire so that charges no longer have a pathway in/out of the metal rod. Once this is done the metal rod has been charged negatively by induction. 

 

Note: had you first removed the positively charged plastic rod and then removed the grounding wire the metal rod would end up neutral. It is only stable for the metal rod to have excess negative charge in the metal rod when the plastic positive rod is nearby. Without the plastic rod nearby the negative metal rod will try to stabilize (i.e. leave) if possible. However if the grounding wire was already removed there is no longer any pathway for charge to enter/leave the metal rod so the excess charge cannot leave. 

 

Example 1:

You are given a small neutral conducting sphere on an insulating stand, a positively charged insulating rod, and a few wires. Additionally, you are near the earth. How do you permanently charge the conducting sphere negative?

Step 1: First, you have to ground the sphere with the wires by connecting the conducting sphere to the ground with it.

Step 2: Then, you bring the positively charged rod closer to the sphere, so that the excess positive charges leave through the wire. Don't touch the rod to the sphere.

Step 3: Finally, you cut the wires so that the sphere is left with negative charges, hence making it a negatively charged sphere.

 

Charging by Friction

 

The third method of charging an object is charging by friction. This is done by rubbing two different materials together in order to transfer charge from one object to the other. One object becomes excess positive while the other becomes excess negative. You might wonder why two neutral objects end up both becoming excess charged since it seems like this would be less stable. The mechanism for this is extremely complicated and not something we will learn in physics this year, but we do know if you rub fur and plastic together one becomes excess positive while the other becomes excess negative. 

 

Tip: either insulator or conductors (under the right circumstances) can be charged through friction. 

 

 

Coulomb's Law & Electric Force

 

We can use Coulomb's Law to calculate the electric force between point charges. "Point charges" means two charges which we can model as a dot. To be able to model the charge as a dot, the distribution of charge within the charged objects needs to be insignificant. As long as the charges are in roughly the shape of a sphere and the size of the sphere is small compared to the distance between the two objects then it's generally ok to model the charges as point charges since the distribution within the sphere is not very important. 

 

where:

Unit of charge = C (stands for Coulomb, kind of analogous to the unit of mass, kg)

(proportionality constant/Coulomb's constant)

 

This version of Colombo's Law does not tell us which way the electric force points. However the direction is not hard to figure out anyway from our Electric Charge Cornerstones. Simply draw a line between the two charges. If they are like charges they repel otherwise they will attract. 

 

Two common errors to avoid:

• Forgetting to square the distance. 
• Using incompatible units. Make sure all distances are in meters and all charges are in Coulombs before you use this rule! 

 

Where is electric charge stored?

Electric charge is stored in atoms. Specifically each proton has the same positive charge and each electron has the same amount of charge except that it is negative. The symbol for the charge of an electron is e^-_. Each electron has a charge of -1.6x10^-19 C. Any excess charge must be an integer multiple of the number of extra electrons (since you can't have 1.5 electrons). So the overall charge Q = ne^-. 

 

One Coulomb of charge is a lot of extra electrons! In particular we can use Q = ne^-  To find how many extra electrons it would take to have 1 C of excess charge. First we know that e^- =-1.6x10^-19 C

 

 

Typically, a Coulomb is too large of an excess charge. Therefore, we would use a micro Coulomb, or μC. μ means x10^-6 so 1 μC = 1x10^-6 C. 

 

Tips:

• Remember that force pairs applicable to all types of force, including Electric Force. Although one object may have more excess charge than another, F_e between them is always the same for both. 
• These problems involve scientific notation because of very large and very small numbers. When solving problems with scientific notation, your final answer must be in the proper format, with 1 digit in front of the decimal point (ex. 9.2 or 8.5), and be times 10 to the appropriate power.
• Example: proper: 5.2 x 10⁵ N. Improper: - 450 x 10³ N. 
• Electrical force, like all forces, is expressed using the unit Newtons.  

 

 

Example 1: 

 

Two charged spheres sit stationary on a table. Both spheres have the same excess positive charge. Draw free body diagrams for each sphere. 

In this case, the two objects have like charges and repel each other. This is why the Fe goes away from each other. Since they are at rest, the forces have to be balanced.    By knowing the type of charge on the two objects, the direction of the force on either one of them can then be determined. In the first diagram left, objects 1 and 2 have like charge causing them to repel each other. As a result, the force on object 1 is directed to the left (away from 2) and the force on object 2 is directed to the right (away from 1).

 

 

Example 2:  

Two objects, A and B, are 20 cm apart. Object A has a charge of 4 μC and object B has a charge of -3 μC. Find the electric force exerted on object A by object B. Be sure to change the cm into m otherwise, your answer will be off. 

 

 

 

Balloon Demo

 

Why do balloons stick to the wall when rubbed on your hair? Why does the balloon eventually fall to the ground?

 

Explanation When you rub a balloon on your hair, you are charging it by friction. Both your hair and the balloon gain opposite amounts of excess charge. In this case we assumed the balloon gained an excess positive charge. In the picture shown left, the right side of the balloon was charged by the hair. Charges cannot move within an insulator so those excess charges are stuck in place.   The wall is a neutral conductor. Therefore the excess positive charges in the wall move away from the positive charges in the balloon because like charges repel. Additionally the excess negative charges in the wall move toward the balloon since opposite charges attract. We know that charges can move freely in the wall since the wall is a conductor.   Even though there are as many attractive interactions as repulsive ones, the opposite charges are closer together then the like excess charges. We know that as the distance between two charges decreases the force increases. Therefore the attractive forces from the opposite charges are stronger then the repulsive forces from the like charges and there is a net attraction.    The balloon and the ball both pull on each other with an equal electric force going in opposite directions. The wall pulls the balloon to the right with an electric force. However the balloon doesn't accelerate into the wall since the wall also compresses and exerts a normal force on the balloon to the left. This normal force causes a vertical friction force which keeps the balloon in place.    Over a long time the balloon will lose it's excess charge since air is a slight conductor. As the net charge of the balloon drops, the electric force between the balloon and wall also decreases. This causes the normal force on the balloon by the wall to decrease since the balloon still does not accelerate. Finally the upwards friction force also drops since it depends on the normal force with the same on/by notation. Once the friction becomes too small the vertical forces are unbalanced and the balloon accelerates downwards.