Exam 1
What does the concept of electric field allow for?
An easier prediction for how charges will interact when there are way too many charges to try and use Coulomb's law for each charge.
Conductor
Conductors are the other main class of materials that DO conduct electricity well and DO allow charge to move freely throughout them
What does C statnd for
Coulombs, the basic unit for charge
When is Gauss's Law especially useful
For calculating electric fields in high symmetry situations where the charge can be enclosed by a simple surface such as a spherical shell, cylinder, or box.
Insulator
Insulators are a main class of materials that in general do NOT conduct electricity well and do not allow charge to move freely throughout them.
What is the more important thing with potential?
Potential difference
What does the electric field strength (magnitude) depend on?
The size of the charge (how much charge) as well as the distance from the charge. The strength decreases quite quickly as distance increases (1/r^2)
What will happen if we place another charge or charges in that region?
They will feel a force.
In specific formula form: U (potential energy)=
U(potential energy)=qV
What is the best way to think about electric potential energy?
if a charge q, moves through a region from one potential to a different potential, i.e. there is a potential difference (V) in that region, the charge will experience a change in energy = qV
Semiconductors
"Semi"conductors are exactly that: they are in between conductors and insulators. They partially conduct electricity well and allow charge to move freely through them.
What is a common connection used to try to figure out what an electric field looks like
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What is a field?
A "field" is simply an idea of the effect that charges have on their surrounding area. It is a way of picturing what the area around a charge looks like in terms of its "electrical effects."
What will a positive and negative charge feel?
A positive charge will feel a force in the same direction as the electric field. Negative charges feel a force in the opposite direction as the electric field.
What does polarization allow?
Allows for a charged object to attract a neutral, uncharged object
How to get the electric field from potential
Basically, if you have a function for the potential in terms of one or more variables: V(x,y,z), simply take the partial derivative with respect to a variable and the result will be the electric field component in that direction. Often this works well because potential is a scalar and is easier to calculate for difficult geometries, then taking a derivative to get electric field becomes much simpler than the direct integration method from Ch 23
How do conductors in equilibrium behave?
Because of the way charges can move in conductors, a conductor in equilibrium automatically becomes an equipotential: all the charge is on the surface, and spreads out uniformly so the entire conductor becomes an equipotential
Why is the electric field useful?
Because there is a direct relation between it and electric force Fe=qE. Meaning that so long as you can establish the overall net electric field direction and magnitude, you can immediately predict how charges will move.
Why are superconductors special?
Charges flow through them with absolutely no losses at all, and therefore they will always flow once the flow is established.
Once we move past point charges, what are we usually interested in?
Comparing the electric potential at two different locations: what becomes important is potential difference or voltage.
Can conductors be charged so that they acquire charge? How?
Conductors can also be charged so that they acquire charge, but it will move around within the conductor so that it spreads out as far away as possible - this means it will remain only on the surface (no charge can reside inside a conductor)
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Each component of the electric field can be calculated by taking the partial derivative of the electric potential function with respect to a particular variable. This is in fact a very nice relationship and is one way to avoid having to deal with the vector nature of electric field directly (since potential is a scalar)
Electric potential
Electric potential due to a point charge is v(pointcharge)=(kq)/r. Since electric potential is only a scalar, if there are multiple charges around, to find the net electric potential you just have to add each up like normal numbers.
What do electrons an protons have in common, in the most basic sense?
Electrons and protons have the same magnitude of charge, which is q=1.602 x 10^-19 C
Electric Field
Eventually, it becomes redundant to keep having to add up all the individual force contributions due to charges. It is bad enough for two or three charges, you can imagine what it is like when there are thousands or millions...So a better overall mathematical picture is needed. This is called the electric field.
Electric Field Lines
Field "lines" are really just the visual representation of what a field would look like if we could actually see it.
How does Polarization occur?
First the charged object, let's call it (+) is brought near the uncharged object. This will establish a reorientation of the atoms and molecules within the uncharged object so that the opposite sign, (-) in this case, is mostly on the surface of the uncharged object. Now the uncharged object has a lot more (-) charge on its surface, the side near the (+) charged object, than in the rest of it. This makes the surface of the uncharged object "like" a charged object, and the two objects will interact - always attract. The objects will always attract because the charged object will always create the opposite sign of charge in the uncharged object (opposites attract).
What is the relationship between electric potential and electric field?
For the nice linear case of two parallel plates separated by a distance = d, the electric field is constant and uniform between the plates. But the electric potential changes as you move from one plate to another. Overall: E(parallel plate)=deltaV/d where deltaV=Ed
Polarization helps to understand what:
How even uncharged objects can interact with charged objects. Charged objects can rearrange the random charge alignment in uncharged objects if the charged object gets close enough. This rearrangement makes the uncharged object like it has a temporary charge near its surface where the charged object is located.
What is the net electric force?
IF there are multiple charges present, there will be multiple forces as well. Since force is a vector, you must add up the forces in vector form in order to determine the overall effect
Various geometries will produce different types of equipotential surface regions, but how will the electric field act?
IT will always be perpendicular to the equipotential surfaces.
How else can Electric Potential be defined?
In terms of an integral as well. Because electric potential energy is related to charge and potential (U = qV), and at the more fundamental level it is associated with work and forces causing displacements, we can make a connection between electric force and electric potential energy to define electric potential
What is the idea of electric flux and Gauss's Law?
It is the most rigorous way of establishing that the only in electrostatics that can create an electric field is charge.
What does Coulomb's law tell us?
It tells us how to quantify the force between charges. The forces between charges are always equal in magnitude, but opposite in direction, just like Newton's Third Law says they must be.
How do we find the electric fields association with a single point charge?
Just divide Coulomb's law (electric force) by the point charge:
Do superconductors allow the charge to flow 100% free?
No, Even good conductors like metals do not allow charge to flow 100% free. The charges do interact slightly with the metal atoms and there is some loss due to collisions and heat.
Is Electric potential a vector?
No, just a scalar.
Gauss's Law
Or put into words: the only thing that can produce an electric field, and therefore electric flux, is charge. Simply determine the total amount of charge enclosed by a surface, and that must be what is responsible for producing an electric flux through that surface.
What is polarization
Polarization is a sort of "re-situating" of charge within an atom or molecule so that more charge of one type (+ or -) is in one region as compared to another. Different materials are more easily "polarized" than others, and in fact there are always people searching for and trying to make materials that are easier and harder to polarize.
What do charges produce regions of positive or negative potential
Positive charges produce regions of "higher" or positive potential, while negative charges produce regions of "lower" or negative potential
What is the movement of charges with potential:
Positive charges will move from high potential to low potential, that is, away from positive charges and toward negative charges. Negative charges will move from low potential to high potential, that is, away from negative charges and toward positive charges
What is Gauss's Law?
Says that electric flux can only be created by charge. So electric flux is equal to the total charge divided by a constant
A particular case we already talked about was for conductors (metals). We said that excess charge cannot exist inside a conductor - it must only be on the surface of the conductor. In terms of Gauss's Law, what does this mean?
Since charge cannot be inside a conductor, then the right side of Gauss's Law will always be zero (for inside the conductor, not the surface), and then that means E = 0 always. So, a conductor cannot have charge inside it, AND it is not possible for an electric field to exist inside a conductor either!
Why can we relate electric potential to electric potential energy?
Since potential is defined as Energy/charge
What do the electric field lines represent?
So for the electric field, these lines represent the direction and magnitude of the electric field in the various parts of a region.
How can we get the electric field from potential?
So we can get electrical potential by integrating the electric field - in a similar vein we can get electric field from potential by taking a derivative. Often this is much easier and again points to the strength and power of thinking about charges/situations in terms of potential instead of electric field
What is important to notice about the nonintegral formula for flux?
So what is important to notice about this version of the definition for flux is that flux is very much dependent on the orientation between the electric field and the surface of interest. And Even though the electric field may be nonzero, the electric flux through a particular surface could still be zero if the surface is oriented such that cos = 90°
Potential Difference:
So, the real quantity of interest is the potential difference between two locations. Basically, how does the potential "here" compare to the potential "over there." If they are not the same, there exists a potential difference and interesting things will happen with charges
Superconductors
The "super" comes from the fact that they can maintain a flow of charge forever once it is established. There is no loss due to heat or other effects in superconductors.
How is the Coulomb force just like any other force?
The Coulomb force is just like any other force: it is a vector so it has magnitude and direction, and it can cause objects (charges) to move. Many times you will combine this new type of force with the ideas from last semester to determine how charges like electrons and protons will move.
What can affect the conductivity of semiconductors?
The conductivity of semiconductors can also be affected by their interaction with light, heat, or other forms of energy. This too is very useful and distinguishes semiconductors from metals.
What is the convention of the electric field of a point charge?
The convention is that electric fields point away in every direction form positive charges and point toward negative charges from every direction. Basically, electric fields start on positive charges and end on negative charges.
What is another way of thinking about the electric field in terms of potential?
The electric field in some region will be the potential difference between two plates divided by the difference between them: E=deltaV/d. This means if you have a ver large potential difference between two places in space and they are not very far apart, there will be a strong electric field in that region.
Where does the idea of electric potential come from?
The fact that if there is a charge in some region, it will produce an electric field in that region.
What will the fore between the two charges be?
The force between two charges is the same magnitude for both charges, but will be in different directions. If attractive, q1 pulls q2 toward it, but if the force is repelling, q1 pushes q2 away from it and vice versa.
How is the electric field defined specifically
The force per unit charge: E=F/q, and has the units of Newtons/Coulomb.
What type of integral is the integral version of flux?
The integral is a surface integral taken over the surface of interest.
If q(inside) = 0 on the right side of Gauss's Law, then the left side must be what?
The left side must also be =0 or E=0
What does Coulomb's law give us?
The magnitude of the electric force.
Just like electric force, electric fields are vectors, what does this imply?
The net electric field due to several charges is the vector sum of each individual field.
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The potential difference can be calculated by looking at the dot product of the electric field and small displacements and then adding them all up along a path from A to B.
Why do we try to move away from Coulomb's law and instead try to focus on electric field?
The real applications. There are way too many charges to account for.
What is the similar direct integration formula for calculating potential that looks much like the electric field version:
The strategy for using this expression to calculate the electric potential is much like for the electric field version. Determine what dq is and also what r is for the particular situation/geometry. Since potential is a scalar, there is no need to worry with vector issues in this formula.
What happens if the charge moves from one potential to another?
Then it experiences a change in potential energy: deltaU=q deltaV.
If a charge =q is located at a place where the potential = V.....
Then it has electric potential energy: U(potential energy)=qV
What does it mean to say that some molecules are already polar on their own?
They are naturally unbalanced with respect to charge and one side has a slight (+) and the other side has a slight (-). Water is an excellent example of a polar molecule.
What is the only difference as far as charge is concerned for electrons and protons?
They are opposite in sign, with electron negative
Why do insulators not carry charge well?
They do not have any free charge
Why do conductors carry charge easily?
They have lots of free charge.
The force can cause charges to move, meaning???
This means that a force is causing a displacement and work is being done by something
How to calculate the electric potential due to a particular charge distribution:
To calculate the electric potential due to a particular charge distribution, the direct integration method can also be used like it was for electric field. It is usually easier in this case because potential is a scalar, but building up the integral is the hardest part
What is the only way to create a potential difference?
To have a separation of charge, which requires that there must be more of one type of charge in one region as compared to some other region.
Why can't we just use Gauss's Law regardless?
To use Gauss's Law, we have to have certain geometric and symmetric situations, but it turns out to be highly useful and much easier than direct integration for some special cases.
How do you find the electric field due to various charge distributions?
Use an integral
Since the electric field is defined as force/charge, then the force that a single charge will experience due to an electric field is what?
Vector Fe=qE
When can work be done?
Work can only be done if there is energy somewhere to do the work, so energy must be involved with these charge situations. There must be a way to look at these ideas in terms of energy.
What is the convention for establishing the direction of the electric field? Is it a vector?
Yes, it is a vector. Electric fields point away from positive charges and point toward negative charges.
Can insulators still be charged?
Yes, meaning that they can acquire a net charge, but it does not move around the material very easiy
Is Electric field a vector? What does that mean?
Yes, so if there are multiple electric fields, you must add them as vectors, which means direction is important!
What should you notice about the equation Fe=qE
a. Both quantities are vectors, so direction is important. b. This time, we are interested in the sign of the charge . c. Positive charges will not change the sign of the equation, so positive charges experience an electric force in the same direction as the electric field. d. Negative charges (electrons) will change the sign of the equation, so negative charges experience an electric force in the opposite direction to the electric field
How do you determine the sign of U in the U(potential energy) formula?
a. Sign of q, must be included, so it can be (+) or (-). b. Sign of V must be included: is the potential diff (+) or (-)? c. If charges are "released," change in PE will always be (-). d. A (+) charge will go from high potential to low potential, so q, will be (+) but V will be (-) because potential diff is defined as: ∆V = Vf − Vi. The overall product qV, will then also be (-): the (+) charge loses PE, but gains it in some other form. e. (-) charge will go from low potential to high potential, so q, will be (-) but V will be (+) because ∆V = Vf − Vi. The overall product qV, will then still be (-): the (-) charge also loses PE. f. When allowed to move freely, charges of any kind will go from high PE to low PE, just like a ball rolling down a hill - the ball doesn't naturally roll uphill b/c uphill is at a higher PE. g. The alternative idea is when charges are not "released" or "let go," but rather are forced to be in some region. In that case, the charges will have to moved against their natural direc, i.e. against the potential, and therefore PE is increased (stored for use later).
How do you use Gauss's Law
a. So first we "enclose" the charge we are interested in with some surface. b. We choose the surface very cleverly to make the math easier - it is best if the electric field is constant in magnitude along the entire surface boundary and the direction of the electric field is the same as the direction of the area vector. c. Enclosing the charge inside a surface will create an electric flux because the field lines produced by the charge(s) will then pass through the surface in some kind of way. d. Gauss's Law says that whatever that total flux is, it must also be equal to the total charge inside the surface divided by a constant ( = permittivity of free space = 8.85 x 10-12 C2/Nm2). e. So long as we can determine the total charge inside the surface and we have some nice symmetry and geometry that makes the integral side of the equation easy, oftentimes calculating the electric field becomes a trivial task.
List 5 rules for making these field lines:
a. Start on positive charges and end (terminate) on negative charges. b. Increase in number where the electric field is strong, therefore becoming more closely spaced. c. Decrease in number where the electric field is weak, therefore becoming more spread out. d. Never cross: if they did, at the intersection they would be in two directions at once, which is not possible! e. The electric field direction at any given point is actually the tangent to the field line at that point. So for straight field lines, it is the same direction. But for curved field lines, it will not be the same general direction (imagine the tangent line to a circle - it does not curve like the circle does).
What is the tricky part in most cases about the integral for the electric field due to an arbitrary charge distribution?
determining what dq is for the particular situation. In general, it will depend on how many dimensions and what coordinate system is best for the particular problem.
Electric Flux
electric "flux" is a concept associated with the following basic idea: how many electric field lines are passing through a surface. It allows for easier deistration of electric fields by using geometry related arguments and tricks
Electric Potential
electric potential is yet another way of describing what will happen when charges are around each other. But, as we will see, potential is perhaps the most useful way of thinking about it.
What is electric potential defined as?
energy/charge. So it is easy to extend the idea of electric potential to electric potential energy.
How do you establish the direction of the electric force?
just remember that opposite charges attract each other and like charges repel each other.
What is the concept of a gravitational field?
objects with mass produce such a field and "disturb" the area around them with respect to gravitational effects.
Any net charge must reside only on where in conductors?
only on the surface. It cannot have a net charge in its interior. In insulators, the net charge can be anywhere.
What is the definition of electric potential?
potential is defined as the energy per unit charge.
Why are semiconductors useful?
sometimes a level of conductivity between that of insulators and conductors is needed. The range of conductivities in semiconductors is very large and can be tuned easily in most cases depending on the desired application. Semiconductors can also be tuned to allow either (+) or (-) charges to dominate their conductivity. In metals, only (-) charges flow (electrons). Being able to have both types of charge flow is VERY useful and in fact allows for most all advanced electronic applications at present.
What does Coulomb's law describe?
the math of the electric force between charges
What is electric flux?
the number of electric field lines passing through a surface. By convention electric flux is positive when electric field lines pass out of a surface and it is negative when field lines pass into a surface.
Interpretation of the definition for potential difference with electric force:
to determine the potential difference between two points in space (A and B), we "add up" the dot product of the electric field in that region with each individual element of the path (1-D) taken to get from A to B. The orientation of the electric field with the displacement direction is important (dot product)
What is potential also called?
voltage
What is the easiest way to determine how much kinetic energy a charge can obtain when it is allowed to move through a potential difference
when the change (loss) in potential energy must be equal to gain in kinetic energy. This is how electrons and protons are accelerated to great speeds in particle accelerators
Equipotential surfaces:
"Equi"potential means just like you'd think from the name: potential is the same. So an equipotential surface is one that has the same potential everywhere on its surface