Introduction to Electromagnetism - 8

Tutorial Make-up Today More on potentials Equipotentials Work (from mechanics) The work, W, done on a system by an agent exerting a constant force on the system is the product of the magnitude, F, of the force, the magnitude Dr of the displacement of the point of application of the force, and cos q, where q is the angle between the force and the displacement vectors Units of Work Work is a scalar quantity The unit of work is a joule (J) 1 joule = 1 newton . 1 meter J = N · m Work Is An Energy Transfer This is important for a system approach to solving a problem If the work is done on a system and it is positive, energy is transferred to the system If the work done on the system is negative, energy is transferred from the system Work Is An Energy Transfer, cont If a system interacts with its environment, this interaction can be described as a transfer of energy across the system boundary This will result in a change in the amount of energy stored in the system Scalar Product of Two Vectors The scalar product of two vectors is written as A . B It is also called the dot product A . B = A B cos q q is the angle between A and B Review scalar product rules Potential Energy Potential energy exists whenever an object which has mass has a position within a force field. The most everyday example of this is the position of objects in the earth's gravitational field. Note that we have an arbitrary 0 reference (the ground) to express U Work and Electric Potential Assume a charge moves in an electric field without any change in its kinetic energy The work performed on the charge is W = ?U = q ?V Think of q=m and ?V=gh Potential Difference in a Uniform Field The equations for electric potential can be simplified if the electric field is uniform: The negative sign indicates that the electric potential at point B is lower than at point A Charged Particle in a Uniform Field, Example A positive charge is released from rest and moves in the direction of the electric field The change in potential is negative The change in potential energy is negative The force and acceleration are in the direction of the field Energy and the Direction of Electric Field When the electric field is directed downward, point B is at a lower potential than point A When a positive test charge moves from A to B, the charge-field system loses potential energy More About Directions A system consisting of a positive charge and an electric field loses electric potential energy when the charge moves in the direction of the field An electric field does work on a positive charge when the charge moves in the direction of the electric field The charged particle gains kinetic energy equal to the potential energy lost by the charge-field system Another example of Conservation of Energy Directions, cont. If qo is negative, then ?U is positive A system consisting of a negative charge and an electric field gains potential energy when the charge moves in the direction of the field In order for a negative charge to move in the direction of the field, an external agent must do positive work on the charge Example of contour representation of heights in a topographic map Equipotentials Point B is at a lower potential than point A Points B and C are at the same potential The name equipotential surface is given to any surface consisting of a continuous distribution of points having the same electric potential In a way, they correspond to contours in a topographic map Potential and Point Charges A positive point charge produces a field directed radially outward The potential difference between points A and B will be Potential and Point Charges, cont. The electric potential is independent of the path between points A and B It is customary to choose a reference potential of V = 0 at rA = ? Then the potential at some point r is Potential cont. Potential plays the same role for charge that pressure does for fluids. If there is a pressure difference between two ends of a pipe filled with fluid, the fluid will flow from the high pressure end towards the lower pressure end. Charges respond to differences in potential in a similar way. Active Figure 25-10 Instructor Franco Gaspari PHY 1040U (Physics for the biosciences) Introduction to Electromagnetism and Optics Lecture 8 February 2, 2006 Fig. 7.2, p.184 Define infinity as the zero reference for the potential energy of a charged particle. The Potential Energy of a particle q in an electric field E is the work done by an EXTERNAL force in bringing the particle from infinity to a point A at coordinates rq By definition The Potential Energy is the same as the work done by the Electrostatic field in taking q from ri to infinity We have 2 charges q1 and q2 at an infinite distance from each other. We move them close to each other at a distance r12 (it does not matter whether we move both or one of them). What is the work done? Since the potential energy is the negative of the work done by the EXTERNAL force, we get: Potential Energy of a system of 2 charges Electric Potential vs. Potential Energy We have an electric field E. We define the Electric Potential V at any point in space as the potential energy per unit charge. V is a function of position. Therefore V is a scalar field. V is independent of the test charge q. It is a property of the original electric field E. Potential Difference O Note: the reference zero and the corresponding potential energy is arbitrary. What really counts is the Potential difference. O Example: what is the work done by the electrostatic field in taking q from to ? q (b) When an object of mass m moves downward in the direction of the gravitational field g, the object–field system loses gravitational potential energy. (a) When the electric field E is directed downward, point B is at a lower electric potential than point A. When a positive test charge moves from point A to point B, the charge–field system loses electric potential energy. A system consisting of a positive charge and an electric field loses electric potential energy when the charge moves in the direction of the field. A system consisting of a negative charge and an electric field gains electric potential energy when the charge moves in the direction of the field. Think always in terms of work done by the field vs. work done by an external force. This can be accomplished by an external force. It is like pulling a spring, when the electron is released it has greater potential energy to move against the field (natural work of the field). The opposite is true for a positive charge. So when considering gain vs. losses consider: Charge (positive or negative) Direction of movement (along the field or against the field) Find the change in the electric potential of an electron when the electrostatic force cause the electron To move vertically upwards for a distance d=520 m 520 m The electric potential of the electron decreases. 2. An ion accelerated through a potential difference of 115 V experiences an increase in kinetic energy of 7.37 × 10–17 J. Calculate the charge on the ion. 4. What potential difference is needed to stop an electron having an initial speed of 4.20 × 105 m/s? 5. A uniform electric field of magnitude 250 V/m is directed in the positive x direction. A +12.0-?C charge moves from the origin to the point (x, y) = (20.0 cm, 50.0 cm). (a) What is the change in the potential energy of the charge–field system? (b) Through what potential difference does the charge move? A uniform electric field of magnitude 325 V/m is directed in the negative y direction in the Figure below. The coordinates of point A are (–0.200, –0.300) m, and those of point B are (0.400, 0.500) m. Calculate the potential difference VB – VA, using the blue path. The potential V(x,y,z) is a function of position and is a scalar quantity. Of course, there are going to be points in space with the same V. If these points are adjacent, they will form what is called an equipotential line or surface. The surface can be imaginary or real. No net work W is done by an electric field on a charged particle when the particle moves between point i and point f on an equipotential surface. work per unit charge The equipotential surfaces produced by a point charge or a spherically symmetrical charge distribution are spheres. For a uniform field, the equipotential surfaces are a family of planes perpendicular to the electric field line