We have studied many forces up until this point which are contact forces. Gravitation and the electric force are not contact forces. Objects do not have to touch for these forces to act. They can act over a distance.

The idea of a field was devised to help explain how these forces can act over a distance. The field refers to the area surrounding an object. In the case of an electric field, the field extends outward from every charge and permeates the all of space. When a second charge is placed in this field, it feels a force because of the electric field at that point.

Electric field

The area around a charged object. This field exerts a force on any charged object in its vicinity. The closer the charged object is brought to the charged object creating the field, the greater the force exerted on it.

An Applet Showing Electric Forces/Fields

Test charge

a positive charge of very small magnitude. The test charge is used to determine the direction of the electric field. The electric field is deined as the force on a test charge with the test charge being so small that it approaches zero. Defining the electric field in this manner means that the electric field only describes the effect of the charges creating the electric field at that point.

Electric field lines

The electric field can be represented with electric field lines. Their density is a measure of the strength of the electric field at that point. Their direction is one that a positive test charge would take in the field. Field lines are always directed from the positive charge and toward a negative charge.

• The fild lines indicate the direction of the electric field; the field points in the direction tangent to the field line at any point.
• Electric field lines start on a positive charge and end on a negative charge.
• The number of lines starting on a positive charge, or ending on a negative charge, is proportional to the magnitude of the charge.
• The closer the lines are drawn together, the stronger the electric field is in that region.
• The field lines between two parallel plates are parallel and equally spaced, except near the edges. Thus the electric field is uniform between two parallel plates (except at the edges).
• The field lines indicate the direction of the electric field.
• Electric field lines never cross. To do so would imply that a positive test charge would have two directions simultaneously at that point.
• Electric field lines are drawn perpendicular to a surface outside of a conductor.

Electric Field mapping

• Locate one or more charge by clicking in the screen.
• The applet will draw the equipotential lines and the electric field lines (which are white).

Locate charges and draw field lines

Electric field strength (or intensity)

symbol is E and SI unit is N/C; the force on a test charge.

Electric field strength of a point charge:

where E is the electric field, k is Coulomb’s constant, and d is the distance between the charge and the test charge. Note: the force depends only upon the magnitude of the point charge producing the field, not on the value of the test charge.

Electric field is a vector quantity; it has both magnitude and direction. The resultant electric field due to several point charges can be determined using the same method as was used in Coulomb's Law problems. Calculate the strength of the electric field due to each point charge at point P. Determine the direction of the electric field by determing the direction that a test charge placed at point P would take. Use vectors to determine the magnitude and direction of the resultant electric field at point P.

Superposition principle for electric fields If the field is due to more than one charge, the resultant field at at point is found by adding the individual fields due to each charge at that point vectorially.

Electric Fields and Conductors

The electric field inside a good conductor is zero when the charges are at rest.

Any net charge on a good conductor is distributed equally on the surface.

The electric field is always perpendicular to the surface outside of a conductor.

Inside a nonconductor (which does not have electrons free to move), an electric field can exist. And, the electric field outside a nonconductor does not necessarily make a perpendicular angle to the surface.

AP Multiple Choice questions:

1. Electric field questions are very common on the AP multiple choice test.
2. You may be asked to describe how the charge is distributed inside, on, or around a charged sphere.
3. You may be asked the magnitude of the electric field inside, on, or around a charged sphere. Or, you may be given several points around, on, or inside a charged sphere and asked where the electric field is the greatest or the least.
4. You may be asked to perform a simple calculation predicting the magnitude of the electric field at some distance from a point charge.
5. You may be given a drawing indicating several distances between or outside of two charged spheres (usually along the x-axis) and asked where the field is zero, the least, the greatest, etc. Remember, E=kq/d² where q is the charge of the point charge. Remember, mentally put a positive test charge at that point to determine the direction of the electric field at that point.
6. You may be asked to perform simple calculations involving the electric field. Remember, F=Eq.
7. You may be asked to predict the magnitude and direction of the electric field between two charged plates. Remember, this field is uniform. At all points between the plates, if edge effects are ignored, the magnitude and the direction are the same.
8. You may be asked to compare the magnitude of the electric field at one location to another location. The charge and/or the distance may be changed.
9. You may be given a diagram with multiple charges in a geometric configuration (triangle or square) and asked to give the direction of the net electric field at point P.
10. They may ask simple questions involving the characteristics of electric field lines.

AP Free Response questions:

1. A very common problem is a charge hanging by a thread in an electric field. You may be asked to calculate the electric field intensity or the charge (depends on what is given). Remember, draw a free-body diagram which only includes forces! The force exerted by the electric field on the charge is given by F=Eq. If hanging from a thread, this acts horizontally. It is offset by the horizontal component of the tension. The vertical component of the tension is offset by the weight of the charge.
2. You may be given a configuration of charges in a triangle or a square. You may be asked where to place an additional charge so that the net electric field on it is zero. Or, you may be asked to calculate the net electric field at point P (and its direction).
3. You may be given two charges along the x-axis and asked to calculate where the electric field is zero due to the two charges.
4. You may be given a charge moving between two charged plates. The charge enters the area between the two plates and initially is moving horizontally. It is very common to be given the voltage. Remember, in a uniform electric field, V=Ed. You can use this to calculate E. Also, remember qV=W. The product of charge and voltage is thus energy. They use this a lot to have you calculate a speed using 1/2mv².

Electric Field Lines

Electric Potential

Two points are said to differ in electric potential if work is done to move a charge from one point to another point in an electric field.

 Potential or Voltage (symbol is V; SI unit is Volt)

a charge an inifinte distance away from a point charge is defined to have a potential of zero. When that charge is moved closer to the point charge, its potential increases or decreases. Its potential is defined to be the work done to move the charge from infinity to that point in the vicinity of the point charge

 Potential Difference (symbol is D V; SI unit is volt)

the potential difference is the work done on a charge to move it from one point with one potential to another point of another potential work done on a charge; or the potential is the potential energy per unit charge. Only differences in potential can be measured.

D V = W / q

In a uniform electric field (a parallel plate capacitor):

V = E d

where E is the electric field strength and d is the separation between the plates in meters

Electric Potential Energy When a charge q moves from point B to point A in an electric field, the change in electric potential energy is simply the negative of the work done to move the same charge from point A to point B. Just as we defined the electric field as the force per unit charge, we will define the electric potential (or potential) as the potential energy per unit charge. If a point charge q has electric potential energy of PE_A at some point A, the electric potential at point A is given by

V_A = PE_A / q
Since only differences in potential are measurable, the potential at point A would simple be the difference in potential energy, or the work done, to move the charge from some point B to point A.
V_BA = V_B - V_A = W_BA/q

Electric potential is a scalar term. When finding the electric potential due to a collection of point charges, you need only add the potentials together with no concern for direction. Include a sign for the potential corresponding to the sign of the charge.

Clarifying the Difference Between Electric Energy and Potential

Determining the Electric Potential at a point The electric potential at a distance r from a single point charge can be derived from the expression for electric field due to a point charge. Also called electric potential of a point charge. The expression for electric potential:

V = k q / r
Notes about Electric Potential:

1. The potential of infinitiy is defined to be zero.
2. If a point charge is positive, the electric potential of the charge is positive. When moving a charge from infinity to this point, the electric potential increases above a zero level.
3. If a point charge is negative, the electric potential of the charge is negative. When moving a charge from infinity to this point, the electric energy decreased below a zero level.
4. To find the electric potential of a configuration of multiple charges when working problems, calculate the separate electric potential of each charge. The electric potential is thus negative if the charge is negative and positive if the charge is positive. Add the electric potentials with these signs to determine the net electric potential at a point.

Electrostatic energy (U) for point charges can be found. It is simply the same thing as "work" in the definition of voltage. Since the electric potential is defined as the potential energy per unit charge, then the change in potential energy of charge q moved between points a and b is simply equal to qV_ab. In other words,

U = q V
If dealing with point charges, U = qV becomes U = k (q₁q₂)/d. If one is trying to find the total electrostatic energy due to a system of charges, one finds the sum of the electrostatice energies between each charge. As in abolute potential, one includes the sign of the charge.

Relationship Between Electric Potential and Electric Field One can describe the effects of charge distribution using either electric field or electtric potential. Electric potential can be easier to use than electric fields because it is a scalar quantity rather than a vector quantity.

1. In a uniform field (such as between two parallel plates), the units for electric field (N/C) can be written as V/m. We find that E = V/d in a uniform field.
2. In a nonuniform field (such as produced by a point charge), the electric field ina given direction at any point in space is equal to the rate at which the electric potential changes over distance in that direction.

Equipotential Lines Just as electric field lines represented the electric field around a charge, equipotential lines represent the electric potential about a charge. In three dimensions, they become equipotential surfaces.

Equipotential Surface An equipotential surface is one on which all points are at the same potential. The potential difference between any two points on an equipotential surface is zero; there is no work done to move a charge between these two points.

An Animation Showing Equipotential Lines

Characteristics of Equipotential Surfaces

1. No work is done to move a charge between two points on the same equipotential surface.
2. Electric field lines are perpendicular to equipotential surfaces.
3. The surface of a conductor is an equipotential surface. (A conductor must be entirely at the same potential in statics. There is no electric field within a conductor in statics because otherwise an electron would experience a force and would move.)

Electron Volt (eV)A unit used to deal with the energy of electrons. One electron volt is defined as the energy acquired by an particle carrying a charge equal to that of the electron as a result of moving through a potential difference of 1 V. It is not an SI unit, just an easier unit to use than Joules sometimes.

1 eV = 1.6 x 10^-19 J

Millikan’s oil drop experiment

accurately measured the charge of an electron

E q = m g

where E is the electric field, q is the charge in Coulombs, and mg is the weight in Newtons

AP Multiple Choice questions:

1. You may be asked to perform a simple calculation determining the electric potential at some point P a distance d from a point charge (remember to use V=kq/d).
2. Remember the definition of potential difference - it is the work done on a charge. If they tell you how much work is done on a certain charge, you can apply this definition to determine the potential difference.
3. They can give you two charges on the x-axis, noting several points either outside or between them, and ask you at which point the electric potential is the greatest, the least, or zero.
4. They may give you charges in a square or triangle and ask you what is the potential of a charge held at point P. (Read notes above on electrostatic energy, U).

AP Free Response questions:

1. They could give you an array of charges (in a square or a triangle) and ask you to determine the electrostatic potential of the array at some point P. They may also ask you to compare the work done to move a charge to this point P compared to another array of charges.
2. A very common type of problem is to have a charge entering the region between two parallel and charged plates of known voltage between them. Again, remember the two important formulas, V=Ed and qV=W. The first is used to calculate E when V and d are known. The second is used to calculate speed when V is known.
3. In this same type of problem, they may ask you to draw the forces on the charge using a free-body diagram. They may ask you to draw the path of the charge between the two plates. They may apply mechanics concepts, asking you to calculate speed, time, or vertical displacement of the charge as it travels between the two plates. Remember, the vertical acceleration is due to the electric field (you can ignor gravity because the charge isn't between the plates long enough for it to have an effect). The horizontal acceleration is zero so the time required to move through the region is found using d=vt.

Sharing of Charge

In a conductor, charges move until all parts of a conductor are at the same potential. If a large and a small sphere have the same total charge, the large sphere will have a lower potential. If a large and a small sphere have the same potential, the large sphere will have the greater charge.

Grounding

the potential of the earth is zero. Any object connected to the earth will have its excess charge flow into the earth. It is considered to be grounded.

Electrostatic charges are only found on the outside of conductors.

Capacitors

Capacitor

a device (sometimes called a condenser) that stores charge in the electric field between its plates. Each plate carries the same amount of charge, one plate being negative and the other being positive. A potential difference exists between the two plates.

Capacitance

symbol is C and SI unit is the Farad, F

q = C V

where q is the charge in Coulombs, C is the capacitance, and V is the potential difference.

Capacitance for a parallel-plate capacitor Capacitance is a proportionality constant. It is a constant for a given capacitor. It does not depend upon charge or voltage. Its value only depends upond the structure and dimensions of the capacitor itself. For a parallel-plate capacitor with plates of area A separated by a distance d of air, the capacitance is given by: This relationship makes sense. Plates with a larger area will have less repulsion between charges (they're further apart) for a given amount of charge q. Thus, more charge can be held. A greater separation means that the charge on each plate exerts less attractive force on the other plate. Less charge is drawn from the battery, and the capacitance is less. Notice the use of the permitivity of free space constant (We learned in our previous unit how Coulomb's constant was related to the permitivity of free space.)

Dielectric  An insulating sheet found in most capacitors between the plates. A dielectric allows higher voltages to be applied without charge crossing the gap. A dielectric allows the plates to be placed closer together without touching, allowing an increased capacitance. A dielectric increases the capacitance by a factor K, which is known as the dielectric constant. For a parallel-plate capacitor, C = K C_o, where C_o is the capacitance without the capacitor.

It requires energy to place charges on the plates of a capacitor. When the capacitor is discharged, this electrical energy is released. The energy stored in a capacitor is equal to the work done to charge it.

Energy = ½ C V² = 1/2 q V

where C is the capacitance, q is the charge, and V is the voltage

When a DC voltage source is connected across an uncharged capacitor, the rate at which the capacitor charges up decreases as time passes. At first, the capacitor is easy to charge because there is little charge on the plates. But, as the charge accumulates, more and more work is needed to move additional charges on the plates because the plates already have charge of the same sign on them. (what happens to the force required to add charge to the capacitor as more and more charge is added?) As a result, the capacitor charges exponentially, quickly at the beginning and more slowly as the capacitor becomes fully charged. In your major lab, your capacitor is fully charged when the difference in potential between its plates is equal to the potential difference supplied by the voltage source. Since potential difference (voltage) is 'work done on a charge,' the maximum work that can be done to move charges on the capacitor is that of the input voltage. At any time, the charge on the plates is given by:

Half-life

The time it takes the capacitor to reach half full of charge is called the half-life and is related to the time capacitive time constant in the following way:

half-life = RC ln 2

where R is resistance in ohms and C is capacitance in Farads

This formula will be used on your major lab to calculate the theoretical value for the charging and discharging half-lifes. The experimental values for the charging and discharging half-lifes will be determined from your graph.

In your major lab, remember that current (symbol is I and SI unit is the Ampere, A) is defined to be a flow of charge. Remember that a resistor (symbol is R and SI unit is the W)) hinders the flow of charge. Initially, when the flow of charge to the capacitor is high, there is a large flow of charge through the resistor. It takes a large amount of work to move the charge across the resistor (remember, voltage is work on a charge). In your major lab, you will be asked why this amount of work done to move a charge across the resistor decreases exponentially with time as the capacitor becomes charged.

Important things to remember about capacitors on the AP test:

1. The electric field between the two charged plates is uniform, the same magnitude and direction at all points, neglecting edge effects.
2. You can use the direction of the electric field to predict the path a charged particle will take between the two plates. For example, if it is an electron and the top plate is positive, it will be deflected upward.
3. Important formulas to remember: V=Ed and qV=W. The first can be used to calculate the electric field intensity between the two plates. The second can be used to determine the increase in kinetic energy of the particle as it passes between the two plates. This allows you to calculate speed.

Capacitors in Series and Parallel Combinations in Circuits

Equivalent CapacitanceThe capacitance of a single capacitor that can be substituted for a combination of capacitors

1. Parallel combination of capacitors:
   • To find the equivalent capacitance, add the individual capacitances.
2. Series combination of capacitors:
   • To find the reciprocal of the equivalent capacitance, add the reciprocals of the individual capacitances.

AP Multiple Choice questions:

1. There are a surprising number of capacitor questions on the AP test.
2. They ask questions in which the separtion between the plates and/or the area of the plates is changed and how that affects charge and/or voltage.
3. They give you two capacitors in parallel (almost always, but sometimes in series) and ask you what is the equivalent capacitance or how much charge is stored in one of the capacitors. Remember, capacitors add in parallel and q=CV.
4. You might be asked to calculate the energy stored in a capacitor.

AP Free Response questions:

1. Not very common except as two charged plates.
2. You could be given a capacitor of known capacitance and voltage and asked to calculate the charge stored on it.
3. As part of the same problem, a dielectric may be inserted between the plates of the capacitor. They will ask you what the potential difference is (same) and what the electric field is (can be calculated using V=Ed).
4. They can ask you for calculations for a capacitor requiring that you know C=kA/d.

AP Objectives-Electric Field

AP Objectives-Electric Potential

AP Objectives-Capacitors & Dielectrics

Electric Field, Electric Potential, and Capacitors Sample Problems

Electric Field, Electric Potential, and Capacitors Homework

Capacitor Major Lab Data Analysis and Questions

AP Electric Forces & Electric Fields Class Problems

AP Electric Potential & Capacitors Class Problems