Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 16: Electric Forces and Fields Electric Charge Conductors & Insulators Coulomb’s Law Electric Field Motion of a Point Charge in a Uniform E-field Conductors in Electrostatic Equilibrium Gauss’s Law

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 §16.1 Electric Charge There are two kinds of electric charge: positive and negative. A body is electrically neutral if the sum of all the charges in a body is zero. Charge is a conserved quantity.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 The elementary unit of charge is e =  C. The charge on the electron is -1e. The charge on the proton is +1e. The charge on the neutron is 0e. Experiments show that likes charges will repel each other and unlike charges will attract each other and that the force decreases with increasing distance between charges.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 An object can become polarized if the charges within it can be separated This body is electrically neutral. By holding a charged rod near the body, it can be polarized

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 §16.2 Conductors and Insulators A conductor is made of material that allows electric charge to move through it easily. An insulator is made of material that does not allow electric charge to move through it easily.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 §16.3 Coulomb’s Law The magnitude of the force between two point charges is: where q 1 and q 2 are the charges and r is the separation between the two charges. and  0 is called the permittivity of free space.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 r q1q1 q2q2 F 12 F 21 r q1q1 q2q2 F 12 F 21 The electric force is directed between the centers of the two point charges. The electric force is an example of a long-range or field force, just like the force of gravity. Attractive force between q 1 and q 2. Repulsive force between q 1 and q 2.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 Example: What is the net force on the charge q 1 due to the other two charges? q 1 = +1.2  C, q 2 =  C, and q 3 =  C. The net force on q 1 is F net = F 21 + F 31 F 31 F 21 

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 The magnitudes of the forces are: Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 Example continued: The components of the net force are: Where from the figure

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 Example continued: The magnitude of the net force is: The direction of the net force is:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 Example (text problem 16.13): What is the ratio of the electric force and gravitational force between a proton and an electron separated by 5.3  m (the radius of a Hydrogen atom)? The ratio is:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 §16.4 The Electric Field Recall : Where g is the strength of the gravitational field. Similarly for electric forces we can define the strength of the electric field E.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 For a point charge of charge Q, the magnitude of the force per unit charge at a distance r (the electric field) is: The electric field at a point in space is found by adding all of the electric fields present. Be careful! The electric field is a vector!

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 Example: Find the electric field at the point P. q 1 = +e x = 0m q 2 = -2e x = 1m P x = 2m E is a vector. What is its direction? Place a positive test charge at the point of interest. The direction of the electric field at the location of the test charge is the same as the direction of the force on the test charge. x

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 q 1 = +e q 2 = -2e P Locate the positive test charge here. q 1 = +e q 2 = -2e P Direction of E due to charge 2 Direction of E due to charge 1 x x Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 The net electric field at point P is: The magnitude of the electric field is: Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 Example continued: The net E-field is directed to the left>

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 Electric field lines Electric field lines are a useful way to indicate what the magnitude and direction of an electric field is in space. Rules: 1.The direction of the E-field is tangent to the field lines at every point in space. 2.The field is strong where there are many field lines and weak where there are few lines. 3.The field lines start on + charges and end on – charges. 4.Field lines do not cross.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Pictorial representation of the rules on the previous slide:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 §16.5 Motion of a Point Charge in a Uniform E-Field A region of space with a uniform electric field containing a particle of charge q (q>0) and mass m.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 FBD for the charge q Apply Newton’s 2 nd Law and solve for the acceleration. FeFe x y One could now use the kinematic equations to solve for distance traveled in a time interval, the velocity at the end of a time interval, etc.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 Example: What electric field strength is needed to keep an electron suspended in the air? FBD for the electron: x y FeFe w To get an upward force on the electron, the electric field must be directed toward the Earth.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 Apply Newton’s 2 nd Law: Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 Example (text problem 16.44): A horizontal beam of electrons moving 4.0  10 7 m/s is deflected vertically by the vertical electric field between two oppositely charged parallel plates. The magnitude of the field is 2.00  10 4 N/C. (a) What is the direction of the field between the plates? From the top plate to the bottom plate

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 (b) What is the charge per unit area on the plates? This is the electric field between two charged plates. Note that E here is independent of the distance from the plates! Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 FBD for an electron in the beam: (c) What is the vertical deflection d of the electrons as they leave the plates? Apply Newton’s 2 nd Law and solve for the acceleration: x y FeFe w Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 What is the vertical position of the electron after it travels a horizontal distance of 2.0 cm? 0 0 Time interval to travel 2.00 cm horizontally Deflection of an electron in the beam Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 §16.6 Conductors in Electrostatic Equilibrium Conductors are easily polarized. These materials have free electrons that are free to move around inside the material. Any charges that are placed on a conductor will arrange themselves in a stable distribution. This stable situation is called electrostatic equilibrium.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 When a conductor is in electrostatic equilibrium, the E-field inside it is zero. Any net charge must reside on the surface of a conductor in electrostatic equilibrium.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 Just outside the surface of a conductor in electrostatic equilibrium the electric field must be perpendicular to the surface. If this were not true, then any surface charge would have a net force acting on it, and the conductor would not be in electrostatic equilibrium.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 Any excess charge on the surface of a conductor will accumulate where the surface is highly curved (i.e. a sharp point).

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 §16.7 Gauss’s Law +Q Enclose a point charge +Q with an imaginary sphere. Here, E-Field lines exit the sphere.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 Look at a small patch of the surface of the imaginary sphere. With a positive charge inside the sphere you would see electric field lines leaving the surface.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 Recall that It is only the component of the electric field that is perpendicular to the surface that exits the surface. E  Surface so that the

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 36 Define a quantity called flux, which is related to the number of field lines that cross a surface: E  This picture defines the value of . Flux > 0 when field lines exit the surface and flux < 0 when field lines enter the surface.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 37 Example (text problem 16.58): Find the electric flux through each side of a cube of edge length a in a uniform electric field of magnitude E. A cube has six sides: The field lines enter one face and exit through another. What is the flux through each of the other four faces?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 38 The flux through the left face is –EA. The flux through the right face is +EA. The net flux through the cube is zero. There is zero electric flux though the other four faces. The electric field lines never enter/exit any of them. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 39 This is Gauss’s Law. The flux through a surface depends on the amount of charge inside the surface. Based on this, the cube in the previous example contained no net charge. Since E  q, the flux through a surface can also be written as

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 40 Summary Properties of Conductors/Insulators Charge Polarization Coulomb’s Law The Electric Field Motion of a Point Charge in an Electric Field Gauss’s Law