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Chapter 16 Electrical Energy and Capacitance Objectives • Electrical potential • Electric Potential from a Point Charge • Electron Volt • Capacitance • Parallel Plate Capacitor • Capacitor Combinations • Dielectrics Potential Difference • Recall that work is done by some force acting for a certain distance – W = Fd • When it comes to electric charges undergoing an electric force from an electric field – F = qE • So work is – W = qEd Electrostatic Force • Because the Coulomb force is the same as the gravitational force, it must also be conservative • So it fits the rules of conservative energies – ΔPE + ΔKE • Solving you see that change in potential is opposite to change in kinetic – ΔKE = -ΔPE • Apply work-kinetic theorem – W = ΔKE – W = -ΔPE Electric Potential • The electric potential is the change in potential energy of a charged object. – Often referred to as a potential difference. • This can vary because of the magnitude of charge. – Potential energy divided by charge • SI unit is Volt –V • 1 V = 1 J/C – denoted by V ΔV = V2-V1 = ΔPE /q = -Ed Electric Potential Between Two Points • Recall that in an electric field, electrons are transferred from positive to negative. • So particles move from positive locations to negative locations. – So a positive charge gains electric potential energy when it is moved in a direction opposite the electric field. • Because it is being pulled away from the “attractive” point, much like lifting a rock off the ground gives it more potential energy. – So a negative charge loses electric potential energy when it moves in a direction opposite the electric field. • Because it is traveling away from the negative center which is what it wants to do anyways. Electric Potential from Point Charge • Every point in space has an electric potential, no matter what charge. • The potential depends on the size of the charge and how far the charge is from the reference point. – Electric potential is a scalar quantity, so direction does not matter. • But the sign does. – So when asked to find the net electric potential, simply find the algebraic sum of the individual potentials. q V = ke r As distance increases, potential decreases Potential Energy Between Points • The potential energy created from those two points depends on the work done to move the charges – Opposite sign charges attract and work is negative – If the work is directly proportional to the separation between the charges. » So if the separation gets smaller, the work is negative. – Meaning the charges give off energy • Same sign produces positive potential energy – Meaning energy added to system PE = -W = q1q2 q2V1 = ke r This shows the electric potential energy due to point 1 created on point 2. Potentials and Conductors • If we needed to find the potential difference for the entire surface of the charged conductor, we must find the work required to move a charged particle through the electrical field. – W = -PE = -qV = - q(V2 – V1) • Keep in mind that no work is required to move a charge when two points have the same electric potential. Properties of a Charged Conductor in Electrostatic Equilibrium • Remembering from Gauss’s Law, any closed object in electrostatic equilibrium has all of its charge gather on its surface. – Thus the electric potential is constant everywhere on the surface. – And the electric potential anywhere inside the object could be close to any point on the surface, so it also has a constant potential inside that is equal to the potential on the surface. • Essentially, an object in electrostatic equilibrium no matter the shape can be thought of a single point charge. Electron Volt • The electron volt is defined as the energy that an electron or proton gains when accelerated through a potential difference of 1 V. – This is measuring energy so the units are in Joules, J. • This concept of electron volts, eV, is most commonly used in atomic and nuclear physics. 1 eV = 1.60 x 10-19 C•V = 1.60 x 10-19 J Equipotential Surfaces • A surface on which all points have the same potential is called an equipotential surface. – No work is required to move a charge at constant speed while on the surface. – The electric field at every point on the surface acts perpendicular to that point on the surface. • This really tells us that no matter the surface characteristics, a diagram can be drawn using each surface as a single point source. Capacitance • We can now set two conducting surfaces, each being a equipotential surface, close enough to each other to create an electric field. • The two surfaces do not have the same potential difference, therefore work can be done between the two. – As the two surfaces are charging by an outside voltage source, electrons are being taken from one surface and transferred to the other surface through the battery. • The charging will stop once the plates reach the same potential difference with each other that the terminals of the voltage source endure. – When the voltage source is removed, the capacitor now becomes the primary voltage source for the circuit. • So capacitance is defined as the ratio of the charge between the conducting surfaces and the potential difference between surfaces. – Denoted by C – Measured in Farads, F • But a Farad is actually a very large number Q C= V – so we typically measure in the range of F to pF. Parallel-Plate Capacitor • The most common design for a capacitor is to place two conducting plates parallel to each other and separated by a small distance. • Distances of millimeters and smaller! • By connecting opposite leads of a power source to each plate, the charges begin to line themselves up according to the potential difference of the battery. – Remember, the capacitor stops charging once it reaches the same voltage as the battery. • Even when the battery is disconnected, the capacitor will maintain the potential difference of the battery until the two plates are again connected by a conducting material. permittivity of free space C = 0 0 = 8.85 x 10-12 C2/(Nm2) A d surface area of one plate separation between the plates Dielectrics • The material between the plates of a parallel plate capacitor can effect the capacitance of the system. • A dielectric is an insulating material that is placed in between plates of a capacitor to increase its capacitance. – Insulators are used because the plates can realign the charges on the surface of the insulator space for the charge to be stored. – That gives the opportunity for more charge to be transferred to the plates of the capacitor for more storage. + + + + + + + + + + - - - - - - - - - - + + + + + + + + + + - - - - - - - - - - Dielectric Constant • Each material is different and has different abilities to give up electrons to help increase the capacitance. – That increase is a multiple factor called the dielectric constant, . • So: C = C0 • This differs from the dielectric strength, which is the largest electric field a capacitor can hold. – There is no relationship between larger the constant, stronger the field. Combinations of Capacitors 1/ Ceq = C1 + C2 • Capacitors can be placed in a parallel orientation such that each plate of the capacitor is exposed to the same potential difference. – • When the circuit is drawn, the branches are parallel to each other and to the voltage source. The potential difference across the capacitors in a parallel circuit are the same. – Thus the equivalent capacitance, Ceq, of a parallel combination of capacitors is equal to the algebraic sum of the capacitances of each individual capacitor. • • Ceq = 1/C1 + 1/C2 Capacitors can be placed in a series orientation such that each capacitor is placed one after another. The potential difference across the capacitors in a series circuit decreases with each capacitor that is passes through. – Thus the equivalent capacitance, Ceq, of a series combination of capacitors is equal to less than any of the individual capacitors. – Do this by adding the reciprocal of each capacitance and setting it equal to the reciprocal of the equivalent capacitance. C1 C1 C2 C2 V V Energy Stored in a Capacitor • Due to the fact that the energy stored in a capacitor is directly related to the work required to transfer that charge from plate to plate, we see the following: – In order for work to be performed, there must be a potential difference between plates in order to carry the charge across. • W = V• Q – Thanks to the Work-Kinetic Energy Theorem, and seeing that Q is the equivalent of mass in the mechanics world • W = ½ Q V – Similar to kinetic energy in the mechanics world – We combine those to produce a series of equations that would help to find the energy stored in a capacitor • PE = ½ Q V = ½ C(V)2 = Q2 / 2C