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Transcript
1/27/17 Electric Fields (2) A. B. Kaye, Ph.D. Associate Professor of Physics 26 January 2017 Tentative Schedule • Yesterday • • History, etc. Definition and conservation of charge • Today • • Coulomb’s Law First homework becomes available (due 2 Feb) • Tomorrow • Electric Fields (continued) ELECTRIC FIELDS Coulomb’s Law 1 1/27/17 Charles-Augustin de Coulomb 14 June 1736 – 23 August 1806 French physicist Major contributions were in areas of electrostatics and magnetism Also investigated in areas of § Strengths of materials § Structural mechanics § Ergonomics Coulomb’s Law • Charles Coulomb measured the magnitudes of electric forces between two small charged spheres • The force is inversely proportional to the square of the separation r between the charges and directed along the line joining them • The force is proportional to the product of the charges, q1 and q2, on the two particles • The electrical force between two stationary point charges is given by Coulomb’s Law Point Charge • The term point charge refers to a particle of zero size (or zero dimensions) that carries an electric charge (positive or negative) • The electrical behavior of electrons and protons is well-described by modeling them as point charges 2 1/27/17 Coulomb’s Law, cont. • The force is attractive if the charges are of opposite sign and repulsive if the charges are of like sign • The force is a conservative force • Mathematically, • The SI unit of charge is the coulomb, C • ke is called the Coulomb constant ke = 8.9875517873681764 x 109 N・m2/C2 • eo is the permittivity of free space • eo = 8.85418782 x 10–12 C2 / N・m2 = 8.85418782 x 10–12 F・m–1 • Coulomb's Law, Notes • Remember the charges (typically) need to be in coulombs e is the smallest unit of charge and is equal to 1.602 x 10–19 C Therefore, 1 C is equivalent to 6.24 x 1018 electrons or protons • • • This can be a relatively large charge – when we charge things by induction, we typically have µC-range charges • This can also be a relatively small charge – a 1 cm3 block of aluminum has ~1023 free electrons in it – roughly 0.1 MC • Remember that force is a vector quantity (hence, the boldface text for the vector quantities in the equation) Vector Nature of Electric Forces • In vector form, • is a unit vector directed from q1 to q2 • The like charges produce a repulsive force between them 3 1/27/17 Vector Nature of Electrical Forces, cont. • Electrical forces obey Newton’s Third Law • That means that the force on q1 is equal in magnitude and opposite in direction to the force on q2 • With like signs for the charges, the product q1・q2 is positive, and the force is repulsive Vector Nature of Electrical Forces, 3 • Two point charges are separated by a distance r. • The unlike charges produce an attractive force between them. • With unlike signs for the charges, the product q1・q2 is negative and the force is attractive. Two Final Notes about Directions 1. The sign of the product of q1・q2 gives the relative direction of the force between q1 and q2 2. The absolute direction is determined by the actual location of the charges 4 1/27/17 Multiple Charges • The resultant force on any one charge (e.g., q1) equals the vector sum of the forces exerted by the other individual charges that are present. • For example, if four charges are present, the resultant force on one of these equals the vector sum of the forces exerted on it by each of the other charges: Zero Resultant Force, Example • From the diagram, find the point at which the resultant force equal to zero • • The magnitudes of the individual forces will be equal Directions will be opposite • The answer should result in a quadratic equation; choose the root that gives the forces in opposite directions Electrical Force with Other Forces, Example • Two identical small charged spheres, each with a mass of 30 g, hang in equilibrium (see sketch). The length L of each string is 0.15 m and the angle q is 5º. • Find the magnitude of the charge on each sphere. 5 1/27/17 ELECTRIC FIELDS A Particle in an Electric Field Electric Field – Introduction • You should have studied the concept of a “force” and “field” in Physics I (e.g., the gravitational force and the field it produces). • Here, will will examine the electric force as a field force. • Remember that field forces can act through space, even if there is no physical contact between objects. Cf. gravity • Electric Field – Definition • An electric field is a vector field that exists in the region of space around a charged object • This charged object is the source charge • This field defines the relationship between every point in space via the Coulomb interaction • When another charged object, the test charge, enters this electric field, an electric force acts on it. This is how we evaluate the effects of the field 6 1/27/17 Electric Field – Definition, cont • The electric field is defined as the electric force on the test charge per unit charge. • The electric field vector, E, at a point in space is defined as the electric force F acting on a positive test charge, qo, placed at that point divided by the test charge: Electric Field, Notes • E is the field produced by some charge or charge distribution, separate from the test charge. • The existence of an electric field is a property of the source charge • The presence of the test charge is not necessary for the field to exist • We can also say that an electric field exists at a point if a test charge at that point experiences an electric force • This means that our test charge serves as a kind of “detector” of the field Electric Field Notes, Final • The direction of E is that of the force on a positive test charge • The SI units of E are N/C 7 1/27/17 Relationship Between F and E § This is valid for a point charge only § One charge (test or source) must be of zero size § For larger objects, the field will likely vary over the size of the object • If q is positive, the force and the field are in the same direction. • If q is negative, the force and the field are in opposite directions. • This is the electric field equivalent of the particle in a gravitational field: Electric Field, Vector Form • Remember Coulomb’s law, between the source and test charges, can be expressed as • Therefore, the electric field will be More About Electric Field Direction • If q is positive, the force is directed away from q: • The direction of the field is also away from the positive source charge: 8 1/27/17 More About Electric Field Direction • If q is negative, the force is directed toward q: • The field is also toward the negative source charge: Electric Fields from Multiple Charges • At any point P, the total electric field due to a group of source charges equals the vector sum of the electric fields of all the charges: Example Given: Four point charges labeled 1 through 4, all with the same magnitude, q, are placed around the origin as shown. Charge 2 is negative; all others are positive. What is the direction of the E-field at the origin? 9 1/27/17 Example 2 Two charges Q1 = +2e and Q2 = –3e are placed as shown in the diagram. What is the x-component of the electric field at the origin? 10