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 –Δ PE + Δ KE Solving you see that change in potential is opposite to change in kinetic –Δ KE = - Δ PE Apply work-kinetic theorem –W = Δ KE –W = - Δ PE

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 Δ V =V 2 -V 1 = Δ PE / 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. V = As distance increases, potential decreases keke q r

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 = q 2 V 1 =keke q1q2q1q2 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 = -q  V = - q(V 2 – V 1 ) 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 CV= 1.60 x 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 –so we typically measure in the range of  F to pF. C = Q VV

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. C = 00 A d  0 = 8.85 x C 2 /(Nm 2 ) separation between the plates surface area of one plate permittivity of free space

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: 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. C = C0C0

Combinations of Capacitors 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, C eq, of a parallel combination of capacitors is equal to the algebraic sum of the capacitances of each individual capacitor. 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, C eq, 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. C1C1 C2C2 V C eq = C 1 + C 2 C1C1 C2C2 V 1 / Ceq = 1 / C1 + 1 / C2

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 = Q 2 / 2C