MODULE 2: ELECTRIC POTENTIAL This module contains the following lessons: Electric potential energy and electric potential GENERAL PHYSICS 2 Quarter III SY 2020-2021 For the Students COVID is still around, but education continues. Everyone is having difficulty at this trying time. But no one has other choice but to cope. Therefore, I urge you to keep yourselves motivated to continue your studies. Hold on because we will surpass this, eventually. This learning packet contains two main lessons – Electric potential energy and Electric potential. I encourage you to read our main reference book, University Physics, as you go through this learning packet. The activities in this module are all group works. May I again remind you NOT TO USE A SINGLE SHEET OF PAPER IN ANSWERING ALL THE ACTIVITIES FOR A LESSON. Use other sheets if necessary. Failure to comply to this simple instruction means a ZERO mark. Should you have any queries or clarifications, kindly send it to the class messenger group chat. Have fun learning through this packet! -Maam Lamb Criteria and Rating 5 4 3 1 0 Physics Concepts Appropriate concepts are clearly stated and employed correctly. Appropriate concepts are clearly stated but employed with errors. Appropriate concepts are identified but not employed. At least one concept identified but not employed. No identified concept. Math Concepts All mathematical steps are clearly shown and they flow easily towards the correct answer. All mathematical steps are shown with minor errors. The mathematical steps are hard to follow. Identifies at least one equation but unable to apply them. Incorrect equations. Answer Correct answer Correct answer analytically but not numerically. Incorrect answer but on the right path. Incorrect answer. No answer Problem Solving Scoring Rubrics https://assessment.fiu.edu/resources/rubrics-and-curriculum-maps/_assets/rubrics/Physics%20Problem%20Solving%20Rubric%20-%20TAMU.pdf LESSON 2: Electric Potential INTRODUCTION At the end of this lesson, you should be able to: 1. Evaluate the potential at any point in a region containing point charges (STEM_GP12EM-IIIc-16); 2. Determine the electric potential function at any point due to highly symmetric continuous- charge distributions (STEM_GP12EM-IIIc-17); 3. Calculate the electric field in the region given a mathematical function describing its potential in a region of space (STEM_GP12EM-IIIc-20); 4. Solve problems involving electric potential (STEM_GP12EM-IIIc-22); 5. Infer the direction and strength of electric field vector, nature of the electric field sources, and electrostatic potential surfaces given the equipotential lines (STEM_GP12EM-IIIc-18). In lesson 1, the concept of electric potential energy has been defined. The potential energy, U, associated with a test charge, q 0, in an electric field has also been discussed. For this lesson, the potential energy per unit charge will be described. Consider figure 1. What do the positive and negative terminals of a battery mean? What does it mean when a battery has 1.5V? In this lesson, the concepts of electric potential and voltage will be discussed. _________________________________________________________________________________________ ELECTRIC POTENTIAL or POTENTIAL DEVELOPMENT It is represented with a capital letter V. It is defined as the potential energy per unit charge associated with a test charge, q0, π at that point. Mathematically, this is expressed as, π = . It is a π0 scalar quantity. Its SI unit is volt (V), where 1V = 1J/C, in honor of Alessandro Volta (figure 2). Relating potential with the work done by an electric force in displacing a point charge from point a to point b, ππ→π = −π₯π is expressed per unit charge at points a and b. Thus, dividing ππ→π = −π₯π by q0, This expression states that the work done by the electric force as q0 moves from point a to b is equal to the difference of the potential at a and potential at b. Potential at a, Va, is Ua/q0 while potential at b, Vb, is Ub/q0. The expression Va – Vb is the potential difference between a and b. It can be written as Vab, which can be read as potential of a with respect to b and is often called Voltage. An instrument that is used to measure potential difference between two points is a voltmeter. Figure 3 shows an analog voltmeter. There are various types of voltmeters available. Potential due to a point charge Recall from lesson the electric potential energy of two-point charges that are interacting. This is π = ππ0 . Since what is emphasized in this part of the lesson is the potential due to a point charge, the electric potential 4ππ0 π π 1 π energy of two-point charges is divided by q0. Then, π = = , where r is the distance from q to the point π0 4ππ0 π at which the potential is evaluated. This expression of potential due to point charge implies that: • Potential is independent to a test charge, q0. • When q is positive, the potential that it produces is positive at all points. • When q is negative, the potential is negative at all points. • When r is infinity, potential is zero. Potential due to a collection of point charges The potential energy associated with a collection of charges, π = π0 π ∑ π, 4ππ0 π ππ results to an expression of potential due to a collection of point charges, expressed as, V = is divided by q0. This π π0 = 1 π π΄ π, where 4ππ0 ππ ri is the distance from the ith charge, qi, to the point at which potential is evaluated. This expression of potential due to a collection of charge implies that the potential is the scalar sum of the potentials due to each charge. Potential due to a continuous distribution of charge If there is continuous distribution of charge along a line, over a surface, or through a volume, the continuous distribution of charge is divided into elements, dq, and the sum of the potentials due to each charge is, 1 β π V= ∫ , where r is the distance from the charge element, dq, to the field point where potential is evaluated. 4ππ0 π This expression implies that when at points where r is infinity, potential of a charge element at that point is zero. Potential difference from an electric field If there is the collection of charge is not known, but the electric field is determined, the potential difference can be expressed in terms of the electric field. The electric force on attest charge is, πΉβ = π0 πΈββ. The work done by this force as q0 moves from a to b is, Wπ→π = ∫ πΉβ ⋅ β πβ = ∫ π0 πΈββ ⋅ β π. This work expression is divided by q0, π€ thus, π£π − π£π = π→π = ∫ π0 πΈββ ⋅ β π = ∫ πΈββ ⋅ β π . This expression implies that the potential difference is π0 independent of the path taken by q0 when it moves from a to b because the electric force is a conservative force. Additionally, this expression also implies that: • When a point charge is positive (figure 4), the electric fields are positive at all points. The electric potential energy decreases as the test charge, q0, moves from point a to b, which results to a decrease in the potential at b. Consequently, Vb is less than Va, thus, potential difference Va – Vb is positive. Figuratively, if the electric field is directed away from the charge, the potential is positive at any finite distance from the charge. If a test charge moves away from the charge in the direction of the electric field, potential decreases. Oppositely, moving in opposite direction of the electric field means potential increases. • When a point charge is negative (figure 5), the electric field created is negative at all points and the potential is negative at any finite distance from the charge. Figuratively, if a test moves in the same direction of the electric field, the potential decreases. Oppositely, moving in opposite direction of the electric field, the test charge moves in increasing potential. • The general rule whether the point charge is positive or negative is, moving with the direction of electric field means moving in the direction of decreasing potential while moving against the direction of electric field means moving in the direction of increasing potential. When the charge of a test charge is known, then the rule is, if a positive test charge experiences a force that is directed with the electric field (from a point charge), it moves toward decreasing potential while a negative test charge experiences a force against the direction of the electric field (from a point charge), it moves toward increasing potential. Example 1. Charge +e = 1.602 x 10-19C moves in the direction of a uniform electric field, E = 1.5 x 107V/m, from point a to b and travelled a total distance of 0.50m. Find the potential difference, Va – Vb. Given: +e = 1.602 x 10-19C; , E = 1.5 x 107V/m; d = 0.50m Solution: π€π→π πΉ ⋅ β ππΈβ ππ − ππ = = = = πΈβ π π π ππ − ππ = πΈβ 7 V (0.50π) ππ − ππ = 1.5π₯ 10 π π½π − π½π = 7.5 x 106V Example 2. An electric dipole, as in figure 6, has charges q1 = +12nC and q2 = -12nC. Find the potentials at a, b, and c. Given: Let r1 be the distance from charge q1 while r2 be the distance from charge q2 At point a: r1 = 0.06m; r2 = 0.04m At point b: r1 = 0.04m; r2 = 0.14m At point c: : r1 = 0.13m; r2 = 0.13m Solution: At point a Va = 1 ππ π΄ 4ππ0 ππ ππ2 12 × 10−9 π −12 × 10−9 π ( + ) π2 0.06π 0.04π π½π = −ππππ½ Vπ = 9π₯109 At point b Vb = Vb = 9π₯109 At point c 1 ππ π΄ 4ππ0 ππ ππ2 12 × 10−9 π −12 × 10−9 π ( + ) π2 0.04π 0.14π π½π = ππππ. πππ½ Vc = 1 ππ π΄ 4ππ0 ππ Vc = 9π₯109 ππ2 12 × 10−9 π −12 × 10−9 π ( + ) π2 0.13π 0.13π π½π = π Example 3. Find the potential at a distance r from a positive point charge, q, as shown in figure 7. Given: point a = r; point b = ∞; dl = dr; θ = 0 (because dr is parallel to E) Solution: π πππ = ππ − ππ = ∫ πΈ β π π πππ = ππ − ππ = ∫ π ∞ πππ = ππ − 0 = ∫ π π½ππ = π 1 π dr 4ππ0 π 2 1 π dr 4ππ0 π 2 π ππ πΊπ π The potential decreases because the test charge moves in the direction of E. Equipotential Surfaces The imaginary graphical representations of electric fields are electric fields, as discussed in module 1. On the other hand, electric potentials are represented by imaginary three-dimensional surfaces called the equipotential surfaces. Such as its name, it represents equal electric potentials at every point. Consider figure 7.1. The red arrows represent the electric field lines, directed away from the charge while the blue curves are the equipotential surfaces. In a region of field lines, equipotential surfaces can be constructed through any point. As can be seen from the figure, there can be no point at two different potentials, therefore, equipotential surfaces for different potentials can never intersect, just like electric fields. If a test charge, q0, is moved over an equipotential surface, its potential energy does not change, consequently, there can be no work done by the electric force on q0. To do work on q0, the field line must be perpendicular to the surface at every point so that he electric force by the field will also be perpendicular to the displacement of the charge moving on the surface. of work done on a test charge in a relatively small displacement. Additionally, as can be observed from the figures 7.1 and 7.2: • There are equal potential differences between adjacent surfaces. • In regions where the magnitude of the electric field is large, the equipotential surfaces are closer together. This is because there is relatively large amount • In regions where the magnitude of the electric field is weak, the equipotential surfaces are farther apart because the magnitude of work done by the electric force is relatively weak as the charge moves farther distances between surfaces. _________________________________________________________________________________________ ENGAGEMENT Activity 2.3. Find the potential at any height y between two oppositely charged plates shown in figure 8. Activity 2.4. Draw equipotential surfaces for figure 5. Draw equipotential surfaces between two negative charges. _________________________________________________________________________________________ ASSIMILATION Activity 2.5. Is there zero net charge when potential difference is zero? Why? Activity 2.6. In figure 1, what does it mean when a battery has 1.5V? _________________________________________________________________________________________ REFERENCE Hugh, Young D. and Roger Freedman. University Physics. 12th edition. 2008
