RI Chap 14 Electric Fields Lecture Notes

14 ELECTRIC FIELDS H2 Physics 9478 Content Page 14.1 Coulomb’s Law 3 14.2 Electric Field Strength 6 14.3 Electric Potential Energy and Electric Potential 12 14.4 Important Relationships 19 14.5 Uniform Electric Fields 24 14.6 Conductors in Electrostatic Equilibrium 30 14.7 Capacitance and Capacitors 34 14.8 Electric Field & Gravitational Field: Comparison & Summary 41 14.9 Appendix 43 Learning Outcomes Candidates should be able to: (a) recall and use coulomb’s law in the form 12 2 0 1 4 QQF rπε= for the electric force between two point charges in free space or air. (b) recall and use 2 0 1 4 QE rπε= for the electric field strength due to a point charge, in free space or air, to solve problems. (c) define electric potential at a point as the work done per unit charge by an external force in bringing a small positive test charge from infinity to that point. (d) use the equation 2 0 1 4 QV rπε= for the electric potential in the field due to a point charge, in free space or air. (e) show an understanding that the electric potential energy of a system of two- point charges is 12 0 1 4 E QQU rπε= . (f) recall that electric field strength at a point is equal to the negative potential gradient at that point and use this to solve problems. (g) calculate the field strength of the uniform electric field between charged parallel plates in terms of the potential difference and plate separation. (h) calculate the force on a charge in a uniform electric field. [in H1 syllabus] (i) describe the effect of a uniform electric field on the motion of charged particles. [in H1 syllabus]

Page | 2 (j) define capacitance as the ratio of charge stored to the potential difference and use C QV= to solve problems. (k) recall that the electric potential energy stored in a capacitor is given by the area under the graph of potential difference against charge stored, and use this and the equations 1 2U QV= , 21 2 Q CU = and 2U CV= to solve problems.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 3 14.1 Coulomb’s Law Charles Coulomb (1736–1806) measured the magnitudes of the electric forces between charged objects using the same torsion balance Cavendish used to meaure the value of the gravitational constant G (see Chapter 8). In 1785, he successfully overcame various obstacles after numerous experimentations and established the inverse- square law for electric force between charges which was analogous to Newton’s law of gravitation. The electric force between two point charges in free space or air is given by Coulomb’s law. https://www.aps.org/publication s/apsnews/201606/physicshisto ry.cfm where F is the magnitude of the electric force between point charges with charges 1Q and 2Q that are separated by a distance r. The constant of proportionality is 1 0(4 )πε − where 12 1 0 8.85 10 F mε −−= × is the permittivity of free space. NOTE • The electric forces that two point charges exert on each other are equal in magnitude and opposite in direction; they constitute an action and reaction pair. • The electric forces are directed along the line joining the two point charges • Both Coulombs’ law and Newton’s law of gravitation are referred to as inverse-square law, due to their 21 r dependence. • The major difference between gravitational force and electric (or electrostatic) force is that the former is always an attractive force while the latter can be attractive or repulsive. This is because there are two types of charges – positive and negative. Like charges repel, while unlike charges attract as shown in Fig.14.1. Fig. 14.1 Forces between point charges Coulomb’s Law Coulomb’s law states that the force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. 12 2 0 1 4 QQF rπε= + r Q1 F Q2 – F Unlike charges attract – + Q1 – F Q2 F + r Like charges repel – F F r – Q1 Q2 –

Page | 4  Using Coulomb’s Law There are two conditions that form the basis of using Coulomb’s law to solve electrostatic problems. 1. Coulomb’s law applies only to point charges. Two charged objects can be modelled as point charges if they are much smaller than the separation between them. 2. Coulomb’s law is a force law and forces are vectors. When more than two charges are present, the resultant force on any of them equals the vector sum of the forces exerted by the various individual charges. Example 1 Two charges Q (+2.0 µC) and q (+1.0 µC) are separated by a distance of 3.0 m. (a) Determine the magnitude and direction of the force acting on q by Q. (b) If q is negatively charged instead, but with the same magnitude, determine the magnitude and direction of the force on q by Q. [Solution] 3.0 m Q q

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 5 Example 2 Three charges Q1 = +2.0 µC, Q2 = +1.0 µC and Q3 = +3.0 µC are placed at the corners of a right-angle triangle with dimensions as shown. Determine the magnitude and direction of the resultant force acting on Q3. 2.0 m Q1 Q3 Q2 1.0 m Problem Solving Strategy 1. Identify point charges or objects to be modelled as point charges. 2. Draw free body diagram to show the force vectors on the identified object of interest. 3. Select an appropriate coordinate system e.g., x - y coordinates. 4. Determine the resultant electric force on selected object by vector summation.

Page | 6 14.2 Electric Field Strength Electric forces like gravitational forces are long- range forces which require no contact for one charged particle to exert a force on another. The concept of a field was developed by Michael Faraday (1791 – 1867) to explain how long- range forces operate without physical contact between interacting objects. A charged particle creates a field of influence around itself that permeates space. Other charged particles present in this field experience a force, which is called an electric force. Electric field strength is a vector quantity. S.I. unit for electric field strength is N C−1 or V m−1. If the electric field strength E at a point in space is known, the electric force F experienced by a charge q placed at that point is given by Electric force is a vector quantity. S.I. unit for electric force is the newton (N). NOTE • The direction of the electric field strength is that of the electric force acting on a small positive test charge placed at that point. • The direction of the electric force on a charged particle depends on whether its charge is positive or negative (Fig. 14.2). A positive charge will experience an electric force in the same direction as the electric field. A negative charge will experience an electric force in the opposite direction to the electric field. • Some A -Level questions refer to electric field strength as just “electric field”. Fig. 14.2 Electric Field An electric field is a region of space in which a charge placed in that region experiences an electric force. Electric Field Strength The electric field strength at a point in an electric field is defined as the electric force exerted per unit positive charge placed at that point. + − F F E E FE q= F qE=

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page | 7 • The formulae above relating the electric force and electric field strength apply to any electric field. • Two types of electric field will be discussed: non-uniform fields due to point charges and uniform fields produced by a pair of parallel charged plates. Example 3 A charge of −2.0 µC experiences a force of 8.0 N when placed a