Chapter 21 Electric Charge and the Electric Field

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1 Chapter 21 Electric Charge and the Electric Field 1 Electric Charge Electrostatics is the study of charges when they are stationery. Figure 1: This is Fig and it shows how negatively charged objects repel each other, and positively charged objects repel each other. Likewise, oppositely charged objects attract each other. 1

3 showing the structure of the atom Figure 3:

2 1.1 Electric Charge and Structure of Matter Figure 2: This is Fig showing the structure of the atom Figure 3: This is Fig showing the structure of neutral and ionized Lithium atoms. 2

3 2 Conductors, Insulators, and Induced Charges Conductors permit electric charge to move easily from one region of the material to another. Insulators inhibit the motion of electric charge from one region of the material to another 2.1 Charging by Induction Figure 4: This is Fig showing the charging of a metal ball by induction. 3 Coulomb s Law The magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This can be written as: F = k q 1 q 2 r 2 (magnitude only) (1) where k = 1/4πɛ o = N m 2 /C 2. The electric permittivity of a vacuum: ɛ o = C 2 /(N m 2 ) 3

4 3.1 The force between two point charges The general rule for the magnitude of the Coulomb force (i.e., the electrostatic force) is: F = k q 1 q 2 r 2 (magnitude only) (2) where a positive result represents repulsion and a negative result represents attraction. 3.2 Charge of the electron and proton q p = q e = e = (35) C the fundamental quantum of charge 3.3 Electrical Force versus Gravitational Force F e = k q 1 q 2 r 2 F g = G m 1 m 2 r 2 (3) Figure 5: This is Fig showing the electrical and gravitational forces between to α particles (i.e., helium nuclei). 4

5 Figure 6: This is Fig showing the forces between two point charges q 1 and q Forces between two point charges N.B. The electrical forces between two stationary point charges is along a straight line joining their positions, either towards each other (i.e., attractive), or away from each other (i.e., repulsive). 5

6 4 Electric Field and Electric Forces A charged body creates an electric field in the space around it. Figure 7: This is Fig showing how the electric field E is determined by measuring the force F o on a test charge q o at various locations around a charged body. Once the electric field E is known, the force on any charge q at that location can be calculated using the equation: F = q E (Force on q in an electric field) (4) This is similar to the equation we used for measuring the gravitational force on any mass m near the surface of the earth: F g = m g (Force on m in a gravitational field) 6

7 4.1 Electric field due to a Point Charge Figure 8: This is Fig showing the electric field vector E produced at the field point P. The electric field E points away from the + charge while the electric field E point toward the charge. E = k q ˆr (5) r2 where ˆr points in the direction from the source S to the field point P. direction of the electric field E is determined solely by the sign of the charge q. The 4.2 Electric-Field Vector due to a Point Charge Figure 9: This is Fig showing electric field E produced by a point charge q at a field point P. 7

4.3 Electron in a Uniform Field Figure 10: This is Fig. 21.20 showing the acceleration of an electron in a uniform electric field.

8 4.3 Electron in a Uniform Field Figure 10: This is Fig showing the acceleration of an electron in a uniform electric field. The electric field in this example is E = N/C. Some of the physical quantities that can be calculated from the above figure include: acceleration velocity kinetic energy time K = 1 2 mv2 8

9 5 Electric Field Calculations Real life is more than a single point-charge. Let s take a look at the electric field produced by multiple point charges, or by a continuous distribution of charges. 5.1 Electric field due to multiple point charges Figure 11: This is Fig showing the electric field E produced by two charges, one positive, the other negative. E = E 1 + E 2 N.B. The picture above illustrates the principle of superposition for multiple electric field vectors. 9

10 5.2 Field due to an Electric Dipole Figure 12: This is Fig showing the electric field E produced by two charges, one positive, the other negative. For an electric dipole, the two charges must be equal and opposite, q 2 = q 1. In this figure, point c is the vertex of an isosceles triangle, and therefore the same distance r from both of the charges. q 1 = 12 nc. E a = 1 4πɛ o ( ) q 1 (0.06m) î + q 2 2 (0.04m) î 2 E c = E 1c + E 2c = 1 4πɛ o r 2 (q 1ˆr 1 + q 2ˆr 2 ) (6) 10

Figure 13: This is a figure I constructed showing the two electric fields created by equal and opposite charges of an electric dipole at a field point P.

11 Figure 13: This is a figure I constructed showing the two electric fields created by equal and opposite charges of an electric dipole at a field point P. The resultant electric field at point P is the superposition of E + and E. where E x and E y are E = E + + E = kq r 2 + ˆr + kq r 2 ˆr = E x î + E y ĵ ( ) x E x = k q (x 2 + (y 1) 2 ) x 3/2 (x 2 + (y + 1) 2 ) 3/2 ( ) (y 1) E y = k q (x 2 + (y 1) 2 ) (y + 1) 3/2 (x 2 + (y + 1) 2 ) 3/2 11

Figure 14: These are the electric field lines for the electric dipole arrangement shown in the previous figure above. This uses the StreamPlot function in Mathematica.

12 Figure 14: These are the electric field lines for the electric dipole arrangement shown in the previous figure above. This uses the StreamPlot function in Mathematica. Notice how the electric field lines start at the positive charge and terminate at the negative charge. The drawing algorithm prevents most of the field lines from starting and terminating on the electric charges for aesthetic reasons. 12

13 5.3 Field due to a ring of charge Figure 15: This is Fig showing the electric field E produced by a uniform ring of charge having a charge per unit length λ = Q/2πa.. E = 1 4πɛ o Q x î (7) (x 2 + a 2 3/2 ) Question: What is the electric field when the field point P is at a distance far away from the charged ring? E = 1 4πɛ o Q x 2 î (8) 13

14 5.4 Field due to a Charged Line Segment Figure 16: This is Fig showing the electric field E produced by a finite line segment of charge Q uniformly distributed over a length 2a. The charge per unit length is λ = Q/2a. E = 1 4πɛ o Q x x 2 + a 2 î (9) Question: What is the electric field if the line segment becomes infinitely long, but λ remains the same? E = λ 2πɛ o x î (10) 14

15 5.5 Field due to a Uniformly Charge Disk Figure 17: This is Fig showing the electric field E produced by a uniform disk of charge where the charge per unit area is σ = Q/πR 2. E x = de x = 1 4πɛ o σ 2ɛ o [ 1 2π σ rx dr (x 2 + r 2 ) 3/2 (11) ] 1 (R2 /x 2 ) + 1 (12) Question: What is the electric field E when the disk becomes very large? E x = σ 2ɛ o (13) 15

16 5.6 Field due to Two Oppositely Charged Infinite Plates Figure 18: This is Fig showing the electric field E inside and outside of two charged plates having uniform but opposite charges. The charge densities of the two plates are +σ and σ, where σ = Q/area. Inside the plates E = σ ɛ o ĵ Outside the plates E = 0 N.B. The electric field vector E is uniform everywhere inside the plates as long as you are not close to the edges. 16

6 Electric Field Lines Figure 19: This is Fig. 21.28 showing the field lines due to three different charge distributions.

17 6 Electric Field Lines Figure 19: This is Fig showing the field lines due to three different charge distributions. The positive charges are the source of the electric field while the negative charges are the sinks for the electric field. 7 Electric Dipoles Electric dipoles are commonly found in atomic and molecular systems. A dipole moment is described by two equal-but-opposite charges +q and q, separated by a distance d. The Dipole Moment is is defined as follows: p = q d (14) where d is the displacement vector pointing from the negative charge to the positive charge. 17

18 7.1 Torque on an Electric Dipole The torque on an electric dipole is defined as follows: τ = r F = p E (15) τ = p E sin φ (16) 7.2 Potential Energy of an Electric Dipole The potential energy for an electric dipole in an electric field is: U = p E (17) U = p E cos φ (18) Figure 20: This is Fig showing the torque on an electric dipole p in a uniform electric field. 18

Physics II (PH2223) Physics for Scientists and Engineers, with Modern Physics, 4th edition, Giancoli

Physics II (PH2223) Physics for Scientists and Engineers, with Modern Physics, 4th edition, Giancoli Topics Covered Electric Charge & Electric Field Electric Potential Capacitance, Dielectric, Electric