SAT Physics Electric Field - Electric Fields Of Point Charges

SAT Physics Electric Field - Electric Fields Of Point Charges

ELECTRIC FIELDS OF POINT CHARGES
Point charges are charges where the electric field is calculated as though it originates from a single point in space. Obvious point charges are individual electrons, protons, and ions. These tiny objects are essentially points in space. For a larger object to be a point charge, it must be spherical in shape and have the charge evenly distributed over its surface or throughout its volume.

Visualizing Uniform Fields
The field of a point charge radiates outward from the center of the charge and appears similar to the gravity field surrounding a planet viewed as a sphere in space. Figure 10.3 depicts the electric fields of individual positive and negative charges. They are not shown interacting with one another.

Figure 10.3. Electrical fields of positive and negative charges

Although the gravity field of a mass always points toward the mass, the direction of the electric field depends on the sign of charge q. Electric fields point away from positive charges and toward negative charges. The density of the lines indicates the strength of the field. The closer the lines are to each other, the stronger the field is.
When two charges are brought near each other, the electric fields intertwine. Figure 10.4 shows the interactions between unlike and like charges

Figure 10.4. Interactions between unlike and like charges

Note that had the like charges on the right side of Figure 10.4 been negative, then the field lines would still have followed the same pattern. However, the arrows would have pointed toward the negative charges.

Magnitude of the Electric Field of Point Charges

The magnitude of the electric field, E, of a point charge is solved with an equation bearing a strong resemblance to the equation of the gravity field, g, at a point in space.

The electrostatic constant, k = 9 x 10⁹ N • m²/C², is analogous to the gravity constant, G. Charge q creates the electric field, while mass m creates the gravity field. The relationship between the charge q and the electric field E is directly proportional. Doubling the size of the charge, q, will double the magnitude of the electric field, E. The distance r is measured from the center of q (or m for gravity) to the point in space where the field is to be calculated. In both of these field formulas, the distance r is inverted and squared. Thus, changes in r are subject to the inverse square law. Doubling the distance r will cause the magnitude of the field to become one-fourth of its original value.
Many problems do not require the entire field to be drawn. Instead, they require a vector direction of the electric field at a specific point in space. This can be accomplished using an imaginary positive test charge. Visualize the test charge at the location where field direction is needed. The direction the test charge would move if released is the same as the direc­tion of the electric field. The force on a positive charge is always in the same direction as the field.

Electric Field of Point Charges

A conducting sphere contains a charge q. The electric field at a set distance from the center of the sphere is E. If the charge on the sphere and the distance from the sphere are both doubled, by what factor would the electric field change?
 
WHAT'S THE TRICK?

A sphere is a point charge, and its electric field is determined by the following equation.

Double the charge, q, and the distance, r, to determine the effect on the electric field, E.

The electric field is reduced by a factor of half.

Superposition of Fields and Force

Problems often involve determining the value of the electric field due to more than one point charge. Each charge creates its own electric field that can be calculated using the following formula.

The formula is solved once for each charge where r is the distance from the charge being calculated to the point where the field is to be determined. This results in several values for the electric field, one for each charge present. Each calculated electric field is a vector that includes a specific direction. All of the electric field vectors can be added together using vector mathematics to determine the total electric field.

 Electric Field Superposition

Distance in meters

A +8 coulomb charge is located 2 meters to the left of the origin. A -4 coulomb charge is located 2 meters to the right of the origin. Determine the magnitude and direction of the electric field at point P located at the origin.
 
WHAT'S THE TRICK?

Identify the +8 C charge as q₁ and the -4 C charge as q₂. Draw and label the electric field vectors, E₁ for q₁ (away from positive charge) and E₂ for q₂ (toward negative charge) at point P.

Determine the magnitudes of E₁ and E₂. Magnitudes of vectors are always positive, so the sign on the charges can be ignored when determining electric field magnitude.

The resultant of adding these vectors is the total electric field due to the superposition of these charges. The direction of these vectors can be seen in the diagram. Since they both point to the right, they can both be regarded as positive vectors.

E = E₁ + E₂ = (18 x 10⁹) + (9 x 10⁹) = 27 x 10⁹ N/C

Superposition

Two negative charges, 4 coulombs and 16 coulombs, are separated by a distance of 1 meter, as shown in the diagram above. Determine the location, as measured from the 4-coulomb charge, where the electric field is zero.
 
WHAT'S THE TRICK?

Both charges create individual electric fields that overlap. In order for their sum to equal zero, the magnitudes of the fields must be equal and their directions must be opposite. A positive test charge can be imagined to the left of, between, and to the right of the charges to locate a possible zero point. Since the charges are negative, the electric field of each charge will point toward that charge. The only place where the electric field vectors point in opposite directions is between the charges, as shown in the diagram below. For the electric field of the smaller charge to equal and cancel that of the larger charge, the zero point must be closer to the smaller charge. An approximate location for the zero point is shown below.

The problem asks for a distance from the 4-coulomb charge. Set this distance as r. The distance from the 16-coulomb charge is then 1 - r. Set the magnitudes of the two fields equal to each other, and solve for the unknown distance r.

Cancel the constant k, and then take the square root of both sides of the equation.

Electric Force Due to Point Charges

The diagrams in Figure 10.5 show the forces acting on point charges.

Figure 10.5. Forces acting on point charges

Although the interaction between two negative charges is not shown, they would repel each other. The diagram would look very similar to that of the two positive charges shown on the left of Figure 10.5.
When two point charges interact, the resulting force acting on each charge can be deter¬mined using Coulomb’s law. Coulomb’s law is extremely similar to Newton’s law of gravity.

The magnitude of the electric force, F_B is determined by multiplying the electrostatic con­stant, k, by the magnitude of the two interacting charges, q₁ and q₂, and dividing this by the distance between the charges squared, r². The magnitude of the electric force is not depen­dent on the sign of the two charges, and the equation can be solved with all positive values.
The positive and negative signs on charges do influence the direction of the electric force, which can be easily determined by looking at the diagram. Like charges repel, while unlike charges attract. Note that the electric field of each charge is not included in Figure 10.5. Including the field arrows for both charges would have created too much clutter. Always remember that the electric force acting on a positive charge will match the direction of the electric field while the force acting on a negative charge is opposite the field direction.
Newton’s third law of motion is always in effect whenever two objects interact. Two force vectors are shown in each diagram in Figure 10.5. The force on charge 1 (object) is created by the electric field of charge 2 (agent). Similarly, the force on charge 2 (object) is created by the electric field of charge 1 (agent). Coulomb’s law solves for the value of both of these force vectors as dictated by Newton’s third law: Whenever two objects interact, there is an equal and opposite force between them.

Superposition of Force
When a system consisting of several point charges is present, force vectors add together in a manner similar to the superposition of electric field vectors described earlier. If three or more charges are present and you must determine the force on one of them due to all the others, use Coulomb’s law and superposition. Use Coulomb’s law to find the force between the charge in question and every other charge acting on it. In addition, you can find the direction of each of these force vectors using the rules for attraction and repulsion. The result will be several force vectors that can be added together using vector addition to determine the net force acting on the charge in question.