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Topic 10: Fields Topic 10: Fields © Kari Eloranta 2015 Jyväskylän Lyseon lukio International Baccalaureate September 8, 2015 © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Introduction to Fields A spherical object creates a radial gravitational field around it (Tsokos 396). A point charge creates a radial electric field around it. Gravitational and electric fields are conservative fields. Conservative Field In a conservative field, the work done by the field on an object, as the object moves from point A to point B in the field, does not depend on the path taken. An important property of conservative fields is that the concepts of potential and potential energy are well defined. In principle, gravitational and electric fields fill the whole universe. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields General Definition of Potential Energy in Conservative Field The energy stored in a conservative field is called potential energy. Only differences in potential energy have physical meaning. Potential Energy As an object moves from point A to point B in a conservative field, the work done by the field on the object W equals the negative of the change in potential energy ∆E pot, that is, W = −∆E pot. (1) © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Renormalising Potential Energy In Topic 10 we redefine potential energy by changing the zero level from the surface of the Earth to infinitely far away from the Earth. As a result, the potential energy becomes negative. The total (mechanical) energy is the sum of kinetic and potential energies. The negative total energy has a precise physical meaning. Negative Total Energy When the total energy of an object in a conservative field is negative, the object is bound to the field. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Introduction to Gravitational Potential Energy Consider an object of mass m initially infinitely far away from the Earth (mass M). When the object moves from infinity to distance r from the centre of the Earth, the gravitational force does work on the object. The gravitational force on the object (mass m ) at distance r from the centre of the Earth is Mm (2) F =G 2 r where M is the mass of the Earth and G is the gravitational constant. Because the gravitational force increases as the object approaches the Earth, the gravitational force is not a constant, but a function of distance (F = F (r )). Because the force is not constant, we cannot calculate the work done by ordinary means. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Work Done by the Gravitational Force By using calculus, we can show (Tsokos page 397) that the work done by the gravitational force on the object, as the object moves from infinity to distance r from the centre of the Earth along any path, is Mm . W =G r (3) By definition, the work done by the field on the object equals the negative of the change in gravitational potential energy ∆E pot W = −∆E pot © Kari Eloranta 2015 Topic 10: Fields (4) Topic 10: Fields Gravitational Potential Energy ∞ = 0), we get If we let the gravitational potential energy at infinity be zero (E pot a new expression for the gravitational potential energy from the work done by the gravitational force W Mm initial final = −∆E pot = −(E pot − E pot ) = −(G r Mm − 0) = −G . r (5) Gravitational Potential Energy The gravitational potential energy of the object at distance r from the centre of the Earth is Mm E pot = −G . (6) r where M is the mass of the Earth, m is the mass of the object, G is the gravitational constant, and r is the distance from the centre of the Earth to the object. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Gravitational Potential Gravitational potential is the gravitational potential energy per unit mass. Gravitational Potential The gravitational potential at distance r from the centre of the Earth is M Vg = −G , (7) r where M is the mass of the Earth, G is the gravitational constant, and r is the distance from the centre of the Earth to the object. The SI-unit of gravitational potential is [E ] [V ] = = J kg−1. [m] The gravitational potential is a scalar quantity. © Kari Eloranta 2015 Topic 10: Fields (8) Topic 10: Fields Gravitational Potential and Field Strength Gravitational Potential The gravitational potential at point P in a gravitational field is the work done per unit mass in bringing a small point mass from infinity to point P. Field Strength g The average gravitational field strength g between points A and B in the field is the negative of the gradient of the potential ∆V g =− ∆r (9) where ∆V is the change in the gravitational potential in moving from point A to point B, and ∆r is the distance across the equipotential lines in the field between A and B. If the gravitational potential as a function of distance is known (Tsokos page 409), the gravitational field strength is the slope of the line in a (r,V ) coordinate system. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Electric Field 10.1 Electric Field Electric charge creates an electric field around a charged object. The simplest type of a charged object is a point charge, which is a very small electrically charged object. In a field line representation, the density of the lines represents the field strength, and the arrowheads represent the direction. 2 2 1 1 0 0 + -1 -1 -2 -2 -3 -3 Figure : A positive point charge creates an outward radial electric field. © Kari Eloranta 2015 − Figure : A negative point charge creates an inward radial electric field. Topic 10: Fields Topic 10: Fields Electric Field #» 10.2 Electric Field Strength E When we place a small positive test charge q to a point in an electric field, the field exerts an electric force on the charge in the direction of the field. The electric field strength is a quantity that gives the magnitude and direction of the electric field at a certain point in the field. Electric Field Strength The electric field strength is the electric force per small positive unit charge. Electric Field Strength The electric field strength is #» F E= q #» #» (10) where F is the electric force on a small positive test charge q at a point in the electric field. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Electric Field #» 10.2 Electric Field Strength E (cont.) The SI derived unit of the electric field strength is N C−1 (newton per coulomb) (or V m−1 (volt per metre), as we study later). Electric field strength is a vector quantity. Direction of Electric Field The direction of the electric field at a point is the direction of the electric force on a small positive test charge at that point. Charge q should be as small as possible, so that it exerts essentially no force on the charges which created the field. That way the test charge does not change the original charge distribution that created the field. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Electric Field 10.2 Electric Field Strength of Point Charge Electric Field Strength of Point Charge The magnitude of the electric field strength at a distance r from a point charge q is q qt F k r2 q E= = =k 2 qt qt r (11) where k = 8.99 × 109 N m2 C−2 is the Coulomb constant, and qt a charge of a small positive test charge. A point charge creates a conservative radial field around itself. In principle, the field extends to infinity, but decreases rapidly as distance to the charge increases. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Electric Field 10.1 Electric Potential Electric Potential The electric potential at point P in an electric field is the work done per unit charge in bringing a small point charge from infinity to point P. 4 4 3 3 2 2 1 1 0 0 + -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 Figure : The equipotential lines are at right angles to the field lines. © Kari Eloranta 2015 − Figure : The density of the equipotential lines increases as the field strength increases. Topic 10: Fields Topic 10: Fields Electric Field 10.2 Electric Potential The electric potential at point P in an electric field is the electric potential energy per unit charge at that point. Electric Potential The electric potential at distance r from the a point charge q Ve = −k , (12) r where q is the charge of the point charge, k is the electric constant, and r is the distance from the point charge to point P. The SI-unit of electric potential is [E ] [V ] = = J C−1. [q] The electric potential is a scalar quantity. © Kari Eloranta 2015 Topic 10: Fields (13) Topic 10: Fields Electric Field 10.2 Electric Potential Difference In a radial field, the equipotential lines are concentric circles around the charge. The density of the field lines decreases in moving in radial direction away from the charge. Only differences in electric potential have physical meaning, not the value of a potential on a point which may be artificially normalised. Electric Potential Difference V When charge q travels from point A to point B in an electric field, the electric potential difference between the points is W ∆Ve = VA − VB = q (14) where W is the work done by the electric field on the charge. Electric potential is a scalar quantity. The unit of electric potential is [E ]/[Q] = J C−1 = 1V (joule/coulomb=volt), © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Electric Field 10.1 Electric Field of Two Long Plates Long, oppositely charged metallic plates produce a constant electric field between them. Notice that the field is not constant near the edges. This is known as the edge effect. + + + + + + + + + + + + −- −- −- −- −- −- −- −- −- −- −- −By definition, the direction of the field is away from the positive charge. © Kari Eloranta 2015 Topic 10: Fields Topic 10: Fields Electric Field 10.2 Accelerating with Electric Fields Consider a small positive test charge in a homogeneous electric field. As the particle moves in the field, the work done by the field on the charge is W = q∆Ve. By the work energy theorem, this is equal to the change in the kinetic energy of the particle ∆E K. Kinetic Energy in Homogeneous Field When an electric field is used to accelerate a charged particle, the kinetic energy of the particle changes. By the work energy theorem, the work done by the field on the particle is W = q∆Ve = ∆E K (15) where q is the charge of the particle, ∆Ve is the change in the electric potential between the starting and final points, and ∆E K is the change in the kinetic energy of the particle. © Kari Eloranta 2015 Topic 10: Fields

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