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Transcript
PHYS105 Work & Energy Review I. Review of Concepts From Physics 104 A. Philosophy In first semester physics, we showed that we could in theory solve ANY problem involving a system of particles moving at speeds less than 10% of the speed of light by determining the net external force acting upon the system and applying Newton’s Laws. However, we found that many problems were difficult to solve by this method because either the individual forces acting upon the system were unknown (collisions) or because of the vector nature of forces (energy problems). Thus, we solved many problems using the scalar concepts of energy, work, and potential energy instead of using the vector force methods directly!! B. Work Definition: Work is the area under a force vs position graph. C. Conservative vs UnConservative Forces If the work done by a force upon a body depends only on the bodies starting and ending point and not upon the path taken then the force is called “conservative” otherwise it is called “non-conservative.” D. Work-Energy Theorem The net work done on a body (i.e. the work done by all forces) is equal to the change in the body’s kinetic energy. This is the key idea involving work and energy. All other formulas come from it!!!! E. Potential Energy – U The negative of the work done by a conservative force as an object is moved from point A to point B is defined to be the difference in the potential energy of the object due to that force between point A and point B. Because potential energy is defined in terms of work (an area under the force vs position graph) only the difference in potential energy has physical meaning!!! Thus, when someone talks about the potential energy at some point or potential energy function, they actually mean the potential energy difference with respect to some implied zero potential reference point!!! C. Gravitational Potential Energy (near the surface of the Earth) – Ug Using the surface of the earth as the zero gravitational potential energy reference point, the gravitational potential energy of an object of mass M at a height of h above the Earth is given by Note: Gravitational potential energy is NOT a scalar field since its value depends on the mass of the object and not just the object’s location. This is true of all types of potential energy. Potential energy is a property of a system of objects. In this case, the gravitational potential energy belongs to system of masses composed of the Earth and the object at height h. B. Example of a Scalar Field We can create a scalar function that depends only upon the object’s location above the Earth dividing the gravitational potential energy function by the mass of the test object. This new function is a scalar gravitational potential field created by the Earth. With this new function, we can easily determine the potential energy stored as object of mass M is moved from the zero potential energy reference location to a point at a height of h above the Earth by multiplying our scalar gravitational potential energy field by the object’s mass. In other words, we can easily find out how much work we must do to move the object from the Earth’s surface to a height h. Although this process may seem unnecessary for the simple problem we are considering here, it is extremely helpful for problems involving more complicated geometry. Furthermore, we will see shortly that a special instrument called the voltmeter has been constructed to calculate the difference in the scalar electric potential between two points in space. II. Electric Potential Energy – UE A. For non-time varying electric fields, the electric force on a test charge q is a conservative force. B. Using the definition of potential energy from Physics 1224, we have that the change in electric potential energy for a test object of charge q as it is moved from point A to point B is given by the equation: Note: The electric potential energy is not a scalar field as it depends on the charge of the test object. The electrical potential energy is a property of a system of charges (those charges that set up the electric field and the test charge). III. Electric Potential Difference (Voltage) - V A. Definition – The electric potential difference between two points in space is defined as the change in the electric potential energy that a test object would experience as it was moved between the two points divided by the charge on the test object. B. Units – Volts C. Electric potential difference depends only on the points in space and not on the test object. Thus, it is a scalar field created by the charges that created the electric field. Electric Potential Difference: (Scalar) Electric Field Vector) D. Electric potential can be uniquely defined at a point in space only after defining a reference point of zero electric potential. For a point charge, we will choose r = as our reference point. IV. Calculating the Electric Potential Difference From the Electric Field A. If you know the electric field, you can calculate the electric potential difference between two points by finding the “negative” of the area under the Electric Field vs Position garph. B. EXAMPLE: What is the electric potential difference between point (x =3) and (x=0) for a constant electric field, E 100. N iˆ C C. Although it is possible to calculate the electric potential given the electric field, this process is rarely done in practices as it defeats the advantage of avoiding the vector math associated with electric fields. Instead, we will follow the same process that we did in PHYS1224 when we calculated the work integral for a few special forces (gravity, spring, etc) to develop potential energy functions. We then used the potential energy functions when solving problems. V. Electric Potential For a Point Charge A. We already know that any problem involving a distribution of charge can be solved by breaking the charge distribution into a series of point charges and then either integrating or summing the individual point charge contributions. B. The electric potential due to a point charge of charge Q at a distance r from the point charge is given by where we have chosen r = as our zero electric potential reference point. VI. Electric Potential For Any Charge Distribution A. Discrete Set of Point Charges The total electric potential for N discrete point charges can be found using our result from part V. The result is VII. Equipotential Surfaces Definition – An equipotential surface is defined as a surface consisting of a continuous distribution of points having the same electric potential (voltage). A. D B C A VBA = VCA = VDA = B. From the definition of electric potential, we know that an equipotential surface is also a constant potential energy surface!!! Thus, in the same way we can walk on the third floor of a building without being hurt by gravity, we can place our hand on the Van de Graaff generator and become charged without being shocked!!! If we step out of a window on the third floor, then gravity will do work on our mass as we move between two surfaces of different gravitational potential energy. If we touch something or someone who is not at the same electric potential as the Van de Graaff then the electric field will do work on the electric charges in our body as they move between the two surfaces of different electric potential. (i.e. You get shocked!!) ** If you are having trouble following this discussion then review the material on the “work energy theorem” and the “conservation of mechanical energy requirement” from PHYS1224 ** C. Electrical Engineering Safety Tip – Electrical engineers and electronic technicians often work on devices that are capable of severely shocking or even killing a person. In order to reduce the possibility of this hazard, they often tie the common (negative) lead to a single reference point on the circuit called “ground.” The electric potential at any point in the circuit is then measured by touching the positive lead of the voltmeter to the desired point while keeping the other hand in your pocket or behind your back. This reduces the chance of you reaching across the circuit and accidentally touching with your hands two points that greatly differ in electric potential. Although the point chosen as the ground can theoretically be any convenient point in the circuit, it is usually specified in the schematics when one is troubleshooting. Electric field is perpindicular to an equipotential surface at every point on the surface. PROOF: Consider two neighboring points on an equipotential surface. The difference in electric potential between the two points must be zero so we have that D. Electric field lines point in the direction of DECREASING electric potential. EXAMPLE 1: Draw the electric field lines and equipotential surfaces for the positive point charge Q shown below: +Q EXAMPLE 2: Draw the electric field lines and equipotential surfaces for the charge distribution shown below: +3Q -Q LAB: We will experimentally determine equipotential surfaces and electric field lines for different charge distributions in lab. Remember to bring graph paper to lab!!! VIII. Conductors and Electric Potential A. All points on the surface of a conductor are at equipotential! We know from previous work that the electric field is perpendicular to the surface of a conductor. From our work in the previous section, it therefore follows that the conductor’s surface is an equipotential surface. B. All points INSIDE a conductor are at equipotential! Surface of Conductor B A Since the electric field inside a conductor is ______________, we have VBA = VB – VA = C. All points in an EMPTY CAVITY inside a conductor are at the SAME electric potential as the CONDUCTOR!! Outer Surface of Conductor B A Cavity VBA = VB – VA =