SAT Physics Electric Field - Charge

SAT Physics Electric Field - Charge

CHARGE
Charge is characteristic of electrons and protons. It is associated with a variety of electric properties. The magnitude of charge on an electron and on a proton is the fundamental value of charge, e = 1.6 x 10^-19 coulombs, where coulombs (C) is the unit of charge. The charge on a proton is positive, while the charge on an electron is negative. When grouped together, the charge on an equal number of protons and electrons will cancel each other. However, if an object contains more protons than electrons or more electrons than protons, the object will have a net charge, q or Q. The variable q is typically used for a small charge, such as a point charge. The variable Q is typically used for a larger charge, such as the charge on plates. Charge, like matter, is conserved. Although the amount of charge remains constant it can move to another location or to another object. Although masses only attract one another, charges are capable of attracting and repelling each other. Opposite charges attract one another, while like charges repel.

Charged Objects
In addition to a electrons and protons, there are other charged objects. If the number of electrons and protons in an atom are not equal, the atom has a net charge and is known as an ion. For example, the sodium ion, Na⁺, has one less electron than a neutral sodium atom. The oxygen ion, O^2-, has two extra electrons compared to a neutral oxygen atom. In addition, objects made of conducting materials, such as metals, have the ability to store excess charges.
        Charged objects are usually split into two categories in beginning physics: point charges and charged plates. Point charges are spherical in nature and include electrons, protons, ions, and charged spheres. Regardless of the size of a charged sphere, the entire charge can be assumed to be located at a point in the center of the sphere. This includes hollow spheres and solid spheres. Charged plates consist of two metal plates, which are typically flat, paral­lel, of equal size, and separated by a distance. The two main types of charged objects create very different electrical effects and employ different equations. The sections that follow will compare and contrast these important charged objects.
      Charge is always found in exact quantities. All charges are made up of either whole num­bers of electrons or of protons. Since the charge on each electron or proton is 1.6 x 10^-19 coulombs, 6.25 x 10¹⁸ electrons or protons total to 1 coulomb of charge.
    A neutral object does not mean the absence of charge. All objects are composed of atoms, which contain electrons and protons. Therefore, all objects contain charge. Neutral objects merely contain the exact same number of electrons and protons, and these opposite charges cancel each other.

Conservation of Charge
Although the individual charge on any one object in a problem may vary, the total charge of all the objects will remain constant. Charges can move from one object to another or can flow through a circuit. However, the total charge of a system (all the objects under examination) at the start of a problem will equal the total charge of the system at the end of a problem. In other words, total charge is conserved.

Charging
Charging an object involves moving extra charges onto or off of the object. How this is accomplished depends on whether the substance to be charged is a conductor or an insulator. A conductor is a substance that holds its electrons loosely. This allows the electrons to move freely throughout the conductor. The best examples of conductors are metals, which are used as wires in electrical circuits to transport electrons. Insulators are substances that hold their electrons tightly. As a result, insulators prevent the motion of charges. Plastics, which are often used to insulate people from electric shock, are a good example of insulators.
Both conductors and insulators can be charged. When a conductor is charged, the excess charges can move throughout the conductor and readily distribute over the entire outer sin- face of the conductor. When an insulator is charged, the charges stay in the spot where they have been placed.
Conduction and induction are the two main methods of charging. Conduction involves physical contact between objects. When conductors touch each other, the excess charges on one conductor can freely flow to another conductor. To charge an insulator, the charges need to be rubbed on the insulator. This is how people build up a static shock on dry days. Charg­ing by induction is done without physically touching the object that will be charged. This is best explained in Example 10.1.

Charging by Induction

Two neutral, uncharged, conducting spheres mounted on insulating stands are in contact with each other. A negatively charged rod is brought near the spheres but does not touch either sphere. How can the spheres both be charged, and what will the sign of the charge be on each sphere?
 
WHAT'S THE TRICK?

Although neutral, the two spheres still contain electrons and protons. Diagrams nor­mally just show excess charges, like those in the rod. The negative charges in the rod will repel the negative charges in the spheres. This will cause the electrons in the spheres to move to the right, making the right sphere negative. The sphere to the left, which now has fewer electrons, will have a positive charge. If the two spheres are now physically separated, each becomes charged. These spheres become charged without coming into contact with the negative rod, which is induction.

ELECTRIC FIELDS
Note that this example also demonstrates conservation of charge. The original charge on the two spheres is zero since the spheres were initially neutral. Although the spheres each become individually charged, the resulting charges are equal and oppo­site. When added together, the total charge on the system (both spheres) is zero.
Charged objects are surrounded by a distortion in space known as the electric field, E. The electric field is similar to the gravity field surrounding masses. Like gravity, the electric field is a vector quantity having both magnitude and direction. The electric field of a charged object creates an electric force, F_e on other charged objects located in the field, just as the gravity field of a mass creates a force on other objects with mass.
Visual representations and the mathematical equations for electric fields and gravity fields are nearly identical. However, charges and their surrounding electric fields vary from mass and gravity fields in some unique ways. Gravity fields always point toward the mass respon­sible for the field. However, electric fields can point either toward or away from the charge depending on the sign of the charge. While gravity fields cause objects only to attract each other, electric fields can cause charged objects to attract or to repel one another. These dif­ferences make electric fields a little more complicated than gravity fields. The direction of the electric field and its effect on positive and negative charges is extremely important.
The SAT Subject Test in Physics involves two common electric field configurations: uni­form electric field between charged plates and the electric field surrounding spherical point charges. Each field type has its own set of equations and unique problem sets.

UNIFORM ELECTRIC FIELDS
Uniform electric fields exist between two parallel plates containing equal but opposite charges. These fields are considered to be uniform since both the magnitude and the direction of the field are the same at all points between the plates.

Visualizing Uniform Fields
Figure 10.1 compares a uniform electric field and a uniform gravity field.

Figure 10.1. Electric field of charged plates (left)
compared to gravity field (right)

Whereas gravity fields always point toward a mass, such as Earth, electric fields are a bit more complicated due to the existence of two types of charge. An electric field points in the same direction as an electric force points when it is acting on a positive charge. To find the electric field at a point in space due to a charge or to a group of charges, imagine a positive test charge at that location. Determine the direction of force on an imaginary test charge placed in that location. This will be the same as the direction of the electric field. This means that electric fields point away from positive charges and toward negative charges. A uniform field is drawn with parallel and equally spaced arrows. Often diagrams on exams consist of the arrows only, and the plates responsible for the electric field are not shown.

Magnitude of Uniform Electric Fields
Often the magnitude of the electric field, E, is given in a problem. Unlike the known value for the gravity field of Earth, g= 9.8 m/s², the electric field is unique to the plates used in each problem. Although the electric field strength, E, may be given in a problem, you may also be required to calculate it using the equations that appear in the following sections. The units of the electric field are newtons per coulomb (N/C). The units of the gravity field will most likely be reported in meters per second squared (m/s²). However when analyzed as a field rather than as acceleration, the units for the gravity field can be reported as newtons per kilogram (N/kg).

Electric Force in Uniform Electric Fields
If a charge, q, such as a proton or an electron, is placed into a uniform electric field, it will experience a electric force, F_E. This is very similar to placing a mass, m, into the gravity field of Earth. See Figure 10.2.

Figure 10.2. Force due to uniform fields

The magnitude of force can be determined using the following equations:

Electricity                     Gravity
F_E = qE                          F_G = mg

A force is an interaction between an object and an agent. The charged plates are the agent creating an electric field, E. Charge q is the object that experiences a force, F_E.The formula for the magnitude of the electric force is not dependent on whether charge q is positive or negative.
Figure 10.2 clearly shows that the direction of the electric force, F_E, is dependent on the sign of the charge located in the field. The force acting on a positive charge, such as a proton, will be in the direction of the electric field. However, the electric force acting on a negative charge is opposite the field.

Kinematics in Uniform Electric Fields
The sum of all forces acting on an object will determine its resulting motion. The electric force is merely another force. Solve problems involving electric force in the same manner as you solve all other problems involving forces.
1. Orient the problem.
2. Determine the type of motion.
3. Sum the force vectors in the relevant direction.
4. Substitute and solve.
One aspect of electric force problems seems to vary from other force problems. Some problems include the force of gravity, while others ignore it entirely. Why and when is gravity important? Compared with electric force, the force of gravity is incredibly weak. Although the force of gravity is present, its effects are often negligible (too small to affect calculations). Electrons, protons, and ions have insignificant mass, and the force of gravity is usually ignored. In order for gravity to matter, the mass of the charged object must be fairly large. As a general rule of thumb, if the object is large enough to be seen, then the force of gravity is probably important.
Problems on exams frequently test the motion of electrons and protons. The direction of the motion depends on the sign of the charge, with like charges moving in opposite directions and unlike charges moving toward one another. When electrons are compared with protons, keep these facts in mind:

• Electrons and protons have the same magnitude of charge.
• When placed into the same electric field, they will both experience the same magnitude of electric force but in opposite directions.
• The electron has less mass, and the same force will cause it to have a greater acceleration than the proton

 Millikan Oil Drop

Robert Millikan determined the charge on an electron by suspending negatively charged oil drops in a uniform electric field created by two charged plates as shown in the diagram below. Determine the charge of an electron in terms of the mass of the oil drop (m), the electric field (E), and the gravity of Earth (g).
 
WHAT'S THE TRICK?

The oil drop has enough mass to include the force of gravity. In addition, there has to be an upward force countering gravity to keep the oil drop stationary.
 
Orient the problem: The electric field points from the positive plate to the negative plate. The electric force on the negative oil drop points upward, opposite the electric field. The force of gravity points downward

Determine the type of motion: This is static equilibrium.
Sum the force vectors in the relevant direction: Since the oil drop is stationary, the sum of the forces is zero. Simply set the two opposing forces equal to each other.

F_E = F_g

Substitute and solve:

qE = mg

Solve q in terms of the other variables.

Motion of Charges in Uniform Fields

When an electron is released from rest in a uniform electric field E, it reaches a velocity of v after traveling a distance of x. In terms of v, what will be the electron’s velocity if the magnitude of the electric field is doubled while traveling the same distance?
 
WHAT'S THE TRICK?

This problem combines force and kinematics for a particle that is accelerating. Gravity, while present, is negligible for particles as small as electrons.
 
Orient the problem: The electric force is the only force included in the calculations. As a result, this problem can be oriented in any manner. 
 
Determine the type of motion: The particle is accelerating. Electrons move opposite the field.Sum the force vectors in the relevant direction:

Substitute and solve:

The mass and charge of an electron remain constant. Acceleration is directly proportional to the electric field. If the electric field doubles, the acceleration will double.

For the speed of an object accelerating from rest, v₀ = 0, and moving a set distance, use the following kinematic equation. Solve for v.

When distance, x,is held constant, velocity is proportional to the square root of acceleration. Doubling acceleration increases the right side of the equation by the square root of 2. To maintain the equality, velocity must also increase by this factor. The new velocity is √2 v.