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Energy Considerations in Electrical Systems 1.) Just as we decided that the force-per-unit-charge E available at a point (whether there was a charge there to experience the force or not) is a more primary and, hence, useful parameter than is a statement of the force on a specific body, the potential-energy-per-unit-charge available at a point (whether there is a charge there to experience the energy or not) is a more useful parameter than is the potential energy of a specific body. Below is a definition, then a summary of how energy is used in electrical systems. The amount of potential-energy-per-unit-charge available at a point is defined as: U VA = A q Called the electrical potential, its units are joules/coulomb, or the volt. Note: This means if you know the voltage at a point and want to know how much potential energy a charge q has at the point, U A = qVA. 1.) Relationships The work a constant force F does on a body as the body moves a distance d: ! ! WF = F • d ! The work-per-unit-charge available between two points due to a constant electric field E: ! ! W F•d ! ! = = E•d q q Note: This means if you know the E field, the displacement (as a vector) and the charge experiencing the work, the work on the charge becomes: ! ! W = qE • d The work a variable force F does on a body as the body moves along a path: ! ! WF = ! F • d r ! The work-per-unit-charge available between two points due to a variable electric field E: ! ! ! W F ! = ! • dr = ! E • dr q q 2.) The work a conservative force with known potential energy function does on a body traveling through the field ! The work-per-unit-charge available between two points due to a static electric field (note that static electric fields generate conservative forces): W "U =! = !"V q q WF = !"U In other words, the change in voltage is related to how much energy a field gives or takes away from a charge moving through the field, as: W = !q"V Combining the two ways to determine work, we get: ! ! F ! • d r = "#U ! Combining the two ways to determine work-per-unitcharge, we get: ! ! E ! • d r = "#V 3.) Notice that the relationship: ! ! ! F • d r = "#U Notice that the relationship: ! ! E ! • d r = "#V gives us a clever way to derive potential-energy functions! gives us a clever way to derive electrical-potential functions! For a constant force and a straight-line path between two points, we can write: ! ! F • d = !"U For a constant electric field and a straight-line path between two points, we can write: ! ! E • d = !"V 4.) So if you are asked to use the conservation of energy, for instance, and you want to know how much potential-energy a charge has when at a point whose electrical-potential (i.e., its voltage) is V, you get the potential-energy by using: VA = UA q ! U A = qVA 5.) Classes of Problems B 1.) For the electric field lines shown to the right: D a.) Which point(s) is/are at higher electrical potential? E-field lines go from higher electrical potential to lower. A C b.) For which point(s) would an electron have a larger potential energy? An electron will travel from higher to lower potential energy. As it will also travel opposite the E-field lines, the large U happens at A or B. c.) Where, if any, are there points with the same electrical potential? A and B, and C and D. d.) How much work would be required for a positive charge to travel from point A to B? None as the points are on an equipotential line. e.) The voltage at A is 8 volts. The electric field magnitude is 12 v/m. The distance between A and C is .7 meters. What is the !voltage at C? ! E • d = !"V A ! = 60 o C 6.) 2.) A 2 coulomb charge is moved through a +6 volt potential difference. a.) How much work is done by the field in the process? W = !q"V b.) Is the field giving energy to the charge or removing energy from the charge? c.) Let’s make the charge equal to -2 coulombs. Assuming the only force acting is that of the field AND the charge was released from rest, 1 how fast will it be going by the end of the run? ( !q ) VA = 2 mv2 + ( !q ) VA 3.) (from Mr White): Two parallel plates are charged to a voltage of 50 volts. a.) If the separation between the plates is .05 meters, what’s the ! ! E • d = !"V electric field strength? b.) What’s the acceleration of an electron in the field? ! ! ( !q ) E = ma 7.) 4.) Derive an expression for the electrical potential generated by a point charge that is positive at some defined point. How will this differ if the charge is negative? 5.) What is the electrical potential of: a.) A sphere that has Q’s worth of charge shot through it uniformly; b.) A sphere with Q’s worth of charge located at its center; c.) Inside a sphere that is a conductor? 6.) Derive an expression for the electrical potential generated by a group of point charges at some defined point. 8.) 7.) For a hoop of radius R with charge Q on it (sketches provided by master White): a.) Derive an expression for the electrical potential generated a distance x units down the hoop’s central axis. b.) Derive an expression for the electric field as it exists at x using the del operator. Go to the file on the del operator. 9.) 7.) For a hoop of radius R with charge Q on it (sketches provided by master White): a.) Derive an expression for the electrical potential generated a distance x units down the hoop’s central axis. A differential bit of charge dq produces a differential electrical potential dV equal to: dq dV = k r dq =k 1/2 2 (x + R2 ) Integrating yields: V = ! dV = =k (x 2 k dq r! Q + R2 ) Note that this is exactly the same field as would have been produced by a point charge a distance r units from the charge. 1/2 10.) 7.) For an insulating solid sphere of radius R and with Q’s worth of charge uniformly shot through the structure (sketch provided by master White): a.) Derive an expression for V(r) where r > R (i.e., outside the structure). b.) Derive an expression for V(r) where R < r (i.e., inside the structure). c.) Derive an expression for the electric field inside the structure. d.) What is the electric field and electrical potential at the center of the sphere? e.) Graph both the electric field and electrical potential field-strengths as a function of r for this system. 11.)